I was reading Stein's book on Complex analysis, and on page 271 they say that every elliptic function $f$ with periods 1, and $\tau$ is a ratio of $\wp$ and $\wp'$. The book doesn't have a proof of this theorem, but it says that this theorem above will be an easy consequence of the following version of it: "Every even elliptic function $F$ with periods 1, $\tau$ is a rational function of $\wp$."
I'm not sure how to get the odd elliptic functions from this: but I think with is that it will maybe be a product of $\wp'$ and some rational function of $\wp$.