TL;DR: There is no problem taking a bath there, unless you run into some private people who think it's a bad idea personally.
Long version: The question is if the well is on public land or not. Let's try to locate it and see:
- The pool the Wikipedia page refers to belongs to a completely different church (St Mary-in-the-Castle Church) and is in a different location.
- The Old St. Helens church grounds are further north. So this is completely unrelated.
- The most likely location of the spring is again somewhere completely different, in a public forest, and not on church land. Let's see if we can find the pilgrims way, the park road and the shining cliff on the map:
Note that the grey areas are completely built over, the green areas are forest. So the first thing we notice is that there is no connection to continue from the park road to the intersection with shining cliff and then continue to reach the pilgrims way, even if you assume that Helens wood and helens park are the same thing. At least on the map. Let's see if we can find on the ground where this goes.
So you need to follow the Helens road, leave the Shining cliff on your right and continue on this path. It's not on the map, but we can see the entrance. There are some paths in the fields ahead of it, but the issue is that wherever they lead, the area of the old church is separated from these fields by a section of housing. So it's somewhere in the forest west of the church. This leaves open doubts if the well even belongs to the church or, as "the source" here said in a later post, that it belongs to one of the old estates in the area and is not at all from 711.
But either way, it's not on today's church ground but rather in a forest with a lot of paths and walkways. The guy who dug out the well some 4-5 years ago
[...] lit candles, burned frankincense and myrrh, buried a rose crystal egg. We also drank the waters, and washed our faces in it too... When it is clear, I'm gonna bathe in it....
He might have his own opinions about if you should be taking a bath in there, too. However I highly doubt that there is any formal, legal or government organization to prevent you from doing so.