There was in fact a Congregational Chapel in Dawley, according to British History Online - Dawley Social and cultural activities:

"The former Congregational chapel off High Street was used as an assembly room in the early 20th century."

And an Institute as well: "A library and reading room in High Street, recorded from 1856, was known by 1870 as the Literary Institute."

And I wouldn't dismiss the mineral water out-of hand - "the strength of Methodism in the area was associated with the growth of a temperance movement in the late 19th century (fn. 38) and there was an Anglican mission to combat drunkenness in the Dawley Bank area in 1882." (And Captain Webb's drowning occurred in 1883.)

Betjeman wouldn't necessarily have worried about whether the details of his picture were precisely in keeping with the details on the ground in Dawley. But I'd not be surprised if he had in fact done a bit of research, because he was interested in that kind of stuff. Not so much to get the poem right, but for the sake of it, and the poem might have come out of that.

More interesting though is the question raised earlier, and not picked up - were there in fact any stories about Captain Webb making a postumous appearance in his native town?