Some of those WH's use 120vac, which is limited in how fast it can reheat the water. Is yours wired with 120 or 240vac.
A typical showerhead uses a bit over 2gpm and not more than 2.5 (by federal regulation). If yours is really old, it could use quite a bit more (or if you took the restrictor out).
Ensure it is plumbed properly - that the hot is going in the hot side. If they're swapped, you'd be drawing water out of the bottom of the tank which is where the coldest water is.