I had never heard of White Line prints until I read your post. A quick google took me to this page which I found interesting.
If you don't want to use watercolors, maybe acrylics with some kind of modifiers in them to either shorten or extend the drying time to what you want might work.
I am assuming you want water soluble inks mainly for the cleanup without having to use solvents? From the reading on that link it sounds like you need an ink or paint that dries on the paper fairly quickly. Is that what you are looking for? Most printmaking inks do not dry very fast.
For what its worth, I had used the regular Speedball inks years ago on some lino block prints and they seemed to work fine. The prints are 20 years old now and they look the same as when I pulled them.
My friend does quite a bit of Sintra block printing with Akua inks, which are, I believe, an oil based ink but they are water soluble for cleanup. Are the Speedball inks you are using their regular block printing inks or the Akua inks that were purchased by and now marketed by Speedball? The Akua inks would most likely take too long to dry for this printmaking method unless you aren't in a hurry or can find a modifier that speeds their dry time to an acceptable amount.