This checks the time required to compute   invariants[10]
(about 6.3 hours with a 2.4 Mega Herz PC). 
It also expresses the corresponding relative invariants  
as differential-polynomial combinations of the basic ones.

The version number of Mathematica used to evaluate this notebook:

$VersionNumber

The main results are summarized on the last few pages.

u[s_, p_, q_, r_] := 
        Floor[ (1/(p + q + r))*
            (s - Sum[(p + q + k)*e[s, p, q, k], {k, r + 1, s - p - q}]
                   - Sum[(p + j + k)*e[s, p, j, k], {j, q + 1, 2}, {k, 0, 
                    s - p - j}] - 
                Sum[(i + j + k)*e[s, i, j, k], {i, p + 1, 2}, {j, i, 2}, {
                    k, 0, s - i - j}])]

nM[s_] := 
        Module[{counter, seq}, counter = 0; 
          seq =  Apply[Sequence, 
              Reverse[Flatten[
                  Table[{e[s, i, j, k], 0, u[s, i, j, k]}, {i, 0, 2}, {j, 
                      Max[1, i], 2}, {k, 0, s - i - j}], 2]]]; 
          Do[If[(
                  Sum[(i + j + k)*e[s, i, j, k], {i, 0, 2}, {j, 
                      Max[1, i], 2}, {k, 0, s - i - j}]) == s, 
              (counter = counter + 1; 
                monM[s, counter] = 
                  Product[
                    D[w[i, j][z], {z, k}]^e[s, i, j, k], {i, 0, 2}, {j, 
                      Max[1, i], 2}, {k, 0, s - i - j}] ), 
              counter = counter], Evaluate[seq]]; counter]

printM[s_] := 
        Do[ Print["monM[", s, ", ", i, "] = ", monM[s, i]], {i, 
            1, nM[s]}]

wPolynomial[s_]:=Sum[x[s, i]*monM[s, i],{i,1,nM[s]}]

sub[s_]:=Flatten[ Table[D[w[i, j][z], {z, k}] -> D[c[i, j][z], {z, k}],{i, 0, 2},{j, Max[1, i], 2},{k,0,s-i-j}] ]

cPoly[s_] := wPolynomial[s] /. sub[s]

m = 2;  c[0, 0][z_] = 1; 
      c[j_, i_][z_] := c[i, j][z] /; j > i;

cOneStar[i_, j_][z_] := 
        Sum[Binomial[m - mu, i - mu]*Binomial[m - nu, j - nu]*
            D[rho[z], {z, i - mu}]*D[rho[z], {z, j - nu}]/rho[z]^2*
            c[mu, nu][z], {mu, 0, i}, {nu, 0, j}];

sub1[s_] :=  
        Flatten[ 
          Table[D[w[i, j][z], {z, k}] -> 
              D[cOneStar[i, j][z], {z, k}], {i, 0, 2}, {j, Max[1, i], 
              2}, {k, 0, s - i - j} ] ]

\(cOneStarPoly[s_] := wPolynomial[s] /. sub1[s]
\)

alpha[0][j_, zet_] = 1; 
      g[zet_] = f''[zet]/f'[zet];

alpha[1][j_, zet_]= g[zet]*Sum[-k, {k, 1, j}];

alpha[2][j_, zet_]=(D[g[zet], zet]*Sum[-k*(k + 1)/2, {k, 1, j}]+g[zet]^2*Sum[(1 + k)^2*k/2, {k, 1, j}] )//Simplify;

cTwoStar[i_, j_][zet_] := 
        Sum[f'[zet]^(mu + nu)*alpha[i - mu][m - i, zet]*
            alpha[j - nu][m - j, zet]*c[mu, nu][f[zet]], {mu, 0, i}, {
            nu, 0, j}]

sub2[s_] := 
        Flatten[ 
          Table[D[w[i, j][z], {z, k}] -> 
              D[cTwoStar[i, j][zet], {zet, k}], {i, 0, 2}, {j, Max[1, i], 
              2}, {k, 0, s - i - j} ] ]

cTwoStarPoly[s_] := wPolynomial[s] /. sub2[s]

firstCondition[s_]:=firstCondition[s] = Expand[rho[z]^s*(cOneStarPoly[s] - cPoly[s])];

secondCondition[s_]:=secondCondition[s] = Expand[f'[zet]^s*(cTwoStarPoly[s] - 
                    f'[zet]^s*(cPoly[s] /. z -> f[zet]))];

v[s_, p_] := 
        Floor[(1/p)*
            (s - Sum[
                  (i + j + k)*e[s, i, j, k], {i, 0, 2}, {j, Max[1, i], 
                    2}, {k, 0, s - i - j}] - Sum[t*h[s, t], {t, p + 1, s}])]

nA[s_]:=Module[{counter, seq1, seq2},counter = 0;seq1 = 
                Apply[Sequence, 
                  Reverse[
                    Flatten[
                      Table[{e[s, i, j, k], 0, u[s, i, j, k]}, {i, 0, 2}, {j, 
                          Max[1, i], 2}, {k, 0, s - i - j}], 2]]];seq2 = 
                Apply[Sequence, 
                  Reverse[Table[{h[s, t], 0, v[s, t]}, {t, 1, s}]]];Do[If[(Sum[
                              (i + j + k)*e[s, i, j, k], {i, 0, 2}, {j, 
                                Max[1, i], 2}, {k, 0, s - i - j}] + 
                            Sum[t*h[s, t], {t, 1, s}]) == s,(counter = counter + 1;monA[s, counter]=Product[
                                D[c[i, j][z], {z, k}]^e[s, i, j, k], {i, 
                                  0, 2}, {j, Max[1, i], 2}, {k, 0, s - i - j}]
                                *rho[z]^(s - Sum[h[s, t], {t, 1, s}])*Product[D[rho[z], {z, t}]^h[s, t],{t, 1, s}]),counter = counter],Evaluate[seq1],Evaluate[seq2]];counter]

printA[s_] := 
        Do[Print["monA[", s, ", ", i, "] = ", monA[s, i]], {i, 1, 
            nA[s]}]

printA[2]

nB[s_]:=Module[{counter, seq1, seq2},counter = 0;seq1 = 
                Apply[Sequence, 
                  Reverse[
                    Flatten[
                      Table[{e[s, i, j, k], 0, u[s, i, j, k]}, {i, 0, 2}, {j, 
                          Max[1, i], 2}, {k, 0, s - i - j}], 2]]];seq2 = 
                Apply[Sequence, 
                  Reverse[Table[{h[s, t], 0, v[s, t]}, {t, 1, s}]]];Do[If[(Sum[
                              (i + j + k)*e[s, i, j, k], {i, 0, 2}, {j, 
                                Max[1, i], 2}, {k, 0, s - i - j}] +  
                            Sum[t*h[s, t], {t, 1, s}]) == s,(counter = counter + 1;monB[s, counter]=Product[
                                (D[c[i, j][z], {z, k}] /. z -> f[zet] )
                                    ^e[s, i, j, k], {i, 0, 2}, {j, 
                                  Max[1, i], 2}, {k, 0, s - i - j}]*f'[zet]^(2 s - Sum[(t + 1)*h[s, t], {t, 1, s}])
                                  *Product[D[f[zet], {zet, t + 1}]^h[s, t],{t, 1, s}]),counter = counter],Evaluate[seq1],Evaluate[seq2]];counter]

printB[s_] := 
        Do[ Print["monB[", s, ", ", i, "] = ", monB[s, i]], {i, 
            1, nB[s]}]

firstCoefficients[s_] := 
        Do[a[s, i] = Coefficient[firstCondition[s], monA[s, i]], {i, 1, 
            nA[s]}]

firstCoefficients[4]

firstZeroCheck[s_] := 
        Expand[firstCondition[s] - Sum[a[s, i]*monA[s, i], {i, 1, nA[s]}]]

secondCoefficients[s_] := 
        Do[ b[s, j] = Coefficient[secondCondition[s], monB[s, j]], {j, 1, 
            nB[s]}]

secondCoefficients[4]

secondZeroCheck[s_] := 
        Expand[secondCondition[s] - Sum[ b[s, j]*monB[s, j], {j, 1, nB[s]}]]

eqs[s_] := 
        Join[Table[a[s, i] == 0, {i, 1, nA[s]}], 
          Table[b[s, j] == 0, {j, 1, nB[s]}]]

rules[s_] := Flatten[Solve[eqs[s], Table[x[s, i], {i, 1, nM[s]}]]]

(*  Print["firstZeroCheck[", s, "] = ", firstZeroCheck[s]];   *)

(* Print["secondZeroCheck[", s, "] = ", secondZeroCheck[s]]; 
         *)

invariants[s_]:=Module[{expr, counter},counter = 0;
		Print["   "];
		Print["   Computation of invariants[", s, "]"]
              ;
		nM[s];Print["Number of Monic Monomials = ", nM[s]];
	nA[s];Print["Number of firstCondition Terms = ", nA[s]];
  nB[s]; Print["Number of secondCondition Terms = ", nB[s]];
		firstCondition[s];  firstCoefficients[s];
		Print["firstZeroCheck[", s, "] = ",  
              firstZeroCheck[s]];
		secondCondition[s]; secondCoefficients[s];
			Print["secondZeroCheck[", s, "] = ", 
              secondZeroCheck[s] ]; 
		eqs[s];    rules[s]; 
		expr = 
                Expand[(Sum[ x[s, i]*monM[s, i], {i, 1, nM[s]}]) /. 
                    rules[s]];Do[ If[Length[Coefficient[expr, x[s, i]]] != 0, 
                  (counter = counter + 1; 
                    Inv[s, counter] = Coefficient[expr, x[s, i]] ),  
                  counter = counter], {i, 1, nM[s]}]; n[s] = counter; Print["n[", s, "] = ",  n[s] ]   ]

Date[]

invariants[10]

Date[]

\(\(\(x[2, 2]\)[z_]\  = \ 
      \((\ \ \(Inv[m, 2, 2]\)[z]\  + \ 
          \((2 \((m - 1)\)/m)\)*\(w[0, 1]\)[z]*\(w[1, 2]\)[z]
			 - \ 
          \((6 \((m - 1)\)/\((5 \((m + 1)\) m)\))\) \(w[0, 2]\)[z]*
            \(w[1, 1]\)[z]
			 + \ \(w[0, 2]\)[z]^2
		 - 
          \((\((5  m + 2)\) \((m - 1)\)^2/\((5 \((m + 1)\) m^2)\))\) 
            \((\(w[0, 1]\)[z])\)^2*\(w[1, 1]\)[z]\ 
		 - \ 
          \((2 \((5  m + 2)\) \((m - 1)\)/\((5 \((m + 1)\) m)\))\) 
            \((\(w[0, 1]\)[z])\)^2*\(w[0, 2]\)[z]\ 
		 + \ 
          \((\((5  m + 2)\) \((m - 1)\)^2/\((5 \((m + 1)\) m^2)\))\) 
            \((\(w[0, 1]\)[z])\)^4\ 
		 + \ 
          \((\((m - 1)\)/3)\)*D[\(w[1, 2]\)[z], z]\ 
		 - \ 
          \((\((m - 1)\)^2/30)\)*D[\(w[1, 1]\)[z], {z, 2}]\ 
		 - 
          \((\((m - 1)\)^2/\((3  m)\))\)*\(w[0, 1]\)[z]*D[\(w[1, 1]\)[z], z]\ 
          
		 - \ 
          \((\((5  m - 4)\) \((m - 1)\)^2/\((15 \((m + 1)\) m)\))\)*
            D[\(w[0, 1]\)[z], z]*\(w[1, 1]\)[z]\ 
		 - \ 
          \((\((m - 1)\)/3)\)*\(w[0, 1]\)[z]*D[\(w[0, 2]\)[z], z]\ 
		 - \ 
          \((\((m - 1)\)/3)\)*D[\(w[0, 1]\)[z], z]*\(w[0, 2]\)[z]\ 
		 + \ 
          \((\((m - 1)\)^2/15)\)*\(w[0, 1]\)[z]*D[\(w[0, 1]\)[z], {z, 2}]\ 

          		 + \ \((\((m - 1)\)^2/15)\)*\((D[\(w[0, 1]\)[z], z])\)^2\ 
		
           + \ \((\((5  m + 2)\) \((m - 1)\)^2/\((5 \((m + 1)\) m)\))\)*
            \((\(w[0, 1]\)[z])\)^2*D[\(w[0, 1]\)[z], z]\ )\); \)\)

\(\(\(x[1, 2]\)[z_]\  = \ 
      \((\ \ \(Inv[m, 1, 2]\)[z]\  + \ 
          \((\((m - 1)\)/m)\)*\(w[0, 1]\)[z]*\(w[1, 1]\)[z]\ 
		 + \ 
          \(w[0, 1]\)[z]*\(w[0, 2]\)[z]\ 
		 - \ 
          \((\((m - 1)\)/m)\) \((\(w[0, 1]\)[z])\)^3
		 + \ 
          \((\((m - 1)\)/4)\)*D[\(w[1, 1]\)[z], z]
			 - 
          \((\((m - 1)\)/2)\) \(w[0, 1]\)[z]*D[\(w[0, 1]\)[z], z]\ )\); \)\)

x[1, 1][z_]  =  
      Inv[m, 1, 1][z]  +  (w[0, 1][z])^2;

\(\(\(x[0, 2]\)[z_]\  = \ 
      \((\ Binomial[m + 1, 3]*\(a[m, 0, 2]\)[z]
		 + \ 
          \((\((m - 1)\)/2)\)*D[\(w[0, 1]\)[z], z]
			 + \ 
          \((\((m - 1)\)/\((2  m)\))\) \((\(w[0, 1]\)[z])\)^2\ )\); \)\)

\(reduceInv[n_, i_]\  := \ 
      Module[{}, \ \ \ 
		\ \ B[n, i, 0]\  = \ Inv[n, i]; 
		\ \ 
        Do[\ B[n, i, k]\  = \ 
            Expand[\ B[n, i, k - 1]\  /. \ 
					\ \ \ \ \ 
                D[\ \(w[2, 2]\)[z], {z, 7 - k}] -> 
                  D[\ \(x[2, 2]\)[z], \ {z, 7 - k}\ ]\ ], \ {k, 1, 7}]; 

        		\ Do[\ 
          B[n, i, 7 + k]\  = \ 
            Expand[\ B[n, i, 7 + k - 1]\  /. 
						\ \ \ 
                D[\ \(w[1, 2]\)[z], \ {z, 9 - k}] -> 
                  D[\ \(x[1, 2]\)[z], \ {z, 9 - k}]\ ], \ {k, 1, 9}]; 
		\ 
        Do[\ B[n, i, 16 + k]\  = \ 
            Expand[\ B[n, i, 16 + k - 1]\  /. 
						\ \ \ 
                D[\ \(w[1, 1]\)[z], {z, 13 - k}] -> 
                  D[\ \(x[1, 1]\)[z], \ {z, 13 - k}]\ ], \ {k, 1, 13}]; 
		
        Do[\ B[n, i, 29 + k]\  = \ 
            Expand[\ B[n, i, 29 + k - 1]\  /. \ 
					\ \ \ \ \ 
                D[\ \(w[0, 2]\)[z], \ {z, 13 - k}] -> 
                  D[\ \(x[0, 2]\)[z], \ {z, 13 - k}]\ ], \ {k, 1, 13}]; 

        		\ \ \ \ \ \ A[n, i]\  = \ B[n, i, 42]; 
        
Print["A[", n, \ ", ", \ i, \ "] = ", 
          \ A[n, i]]\ ]\)

Do[ reduceInv[10, i] ,    {i, 1, n[10]} ]

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MemoryInUse[]