There is also information in these three earlier discussions (some of it repeated here):

Tom O'Bedlam's Song
Info Req:Tom of Bedlam
Lyr/Chords Req: Bedlam Boys

The institution is still called Bethlem Royal Hospital; or at least it was in the 1960s and early '70s, when I lived a mile or so from it; I don't recall anyone who didn't work there ever calling it Bedlam, or even, I think, making the connection.  It was just an ordinary psychiatric hospital by then, of course.  Most of the patients were voluntary, and some of them would turn up in the local pub from time to time; I only once recall any trouble, and that was mostly the fault of one of the locals.

I knew one of the administrators, and he used to regale us with stories about his regular late-night trips to the woods to dump all the embarrassing things that the kleptomaniacs (his word) had pinched from places unknown.  And there was the time when a patient held an entire ward at gunpoint for some while -until they realised that his gun was actually made out of soap.  It had apparantly been a very realistic model, though; the man had talent.

Tame stuff, compared to the old days, and just as well.