One possibility is your phone may have background apps trying to work or synchronise. This has the possibility of crippling your internet speed.
I can't say I've ever had this happen on mobile, but it has happened to me on PC, where a simple uninstall for a certain program resolved everything.
If there are any apps running in the background, I'd terminate them and then connect to Wi-Fi. If the speed's good again, I'd try re-using everything else one app at a time until the culprit's identified.
Wish I could be of more help.