If you don't have the original casewise data but know the correlations (and hopefully the variances and the sample size) you may simply generate random data having those correlations and analyze that dataset as usual by the canonical correlations program that take in raw data. This way, every output will be correct except the computation of canonical variates' values - for this would need the true data you don't have.
But anyway, if you want to program canonical correlation analysis youself, here is a step-by-step algorithm for you. You may use any language having basic linear algebra matrix functions.
Let $\bf R_1$ be correlations in Set1 of $p_1$ variables. $\bf R_2$ be correlations in Set2 of $p_2$ variables. $\bf R_{12}$ be $p_1 \times p_2$ correlations between the sets.
Also, normally you have to know variances of all the variables. Make $\bf S_1$ the diagonal matrix containing standard deviations in Set1; likewise $\bf S_2$ the diagonal matrix with standard deviations in Set2. If you don't know the variances, assume that they all = 1. Then, unstandardized canonical coefficients will be equal to the standardized ones.
Find $\bf H_1$, the Cholesky root of $\bf R_1$: an upper-triangular matrix whereby $\bf{H_1'H_1=R_1}$. (Please note that in the Wikipedia they show it transposed, as "L", lower-triangular.) Likewise, find $\bf H_2$, the Cholesky root of $\bf R_2$.
Compute $\bf W$:
$\bf = {H_1'}^{-1} R_{12} {H_2}^{-1}$, if $p_1 \le p_2$; or
$\bf = {H_2'}^{-1} R_{12}' {H_1}^{-1}$, if $p_1 \gt p_2$.
Do singular-value decomposition of $\bf W$, whereby $\bf W=UDV'$.
Canonical correlations $\gamma_1, \gamma_2,...,\gamma_m$ where $m=\min(p_1,p_2)$ stand on the diagonal of $\bf D$. How to test them for significance - see here.
Compute standardized canonical coefficients $\bf K_1$ (for Set1) and $\bf K_2$ (for Set2):
$\bf K_1 = H_1^{-1}U$ and $\bf K_2 = H_2^{-1}V$ (first $p_1$ columns of $\bf K_2$), if $p_1 \le p_2$; or
$\bf K_1 = H_1^{-1}V$ (first $p_2$ columns of $\bf K_1$) and $\bf K_2 = H_2^{-1}U$, if $p_1 \gt p_2$.
Compute unstandardized canonical coefficients $\bf C_1$ (for Set1) and $\bf C_2$ (for Set2):
$\bf C_1 = S_1^{-1}K_1$ and $\bf C_2 = S_2^{-1}K_2$.
Compute canonical loadings $\bf A_1$ (for Set1) and $\bf A_2$ (for Set2):
$\bf A_1 = S_1^{-1}(S_1R_1S_1)C_1$ and $\bf A_2 = S_2^{-1}(S_2R_2S_2)C_2$ .
Mean squares in columns of $\bf A_1$ are the proportion-of-variance in Set1 explained by its own canonical variates. Likewise, analogously in $\bf A_2$.
Compute canonical cross-loadings $\bf A_{12}$ (for Set1) and $\bf A_{21}$ (for Set2):
$\bf A_{12} = S_1^{-1}(S_1R_{12}S_2)C_2$ and $\bf A_{21} = S_2^{-1}(S_1R_{12}S_2)'C_1$ .
Mean squares in columns of $\bf A_{12}$ are the proportion-of-variance in Set1 explained by the opposite set's canonical variates. Likewise, analogously in $\bf A_{21}$.
Compute canonical variates scores (if you have casewise data at hand):
Variates extracted from Set1 $\bf Z_1=X_1K_1$ and variates extracted from Set2 $\bf Z_2=X_2K_2$, where $\bf X_1$ and $\bf X_2$ are the (centered) variables of Set1 and Set2.
The variates are produced standardized (mean = 0, st. dev. = 1). Pearson correlation between variates $\bf Z_{1(j)}$$Z_{1(j)}$ and $\bf Z_{2(j)}$$Z_{2(j)}$ is the canonical correlation $\gamma_j$. For visual explanation of the idea of canonical correlations please look in here.
