... why, on the Northern line's city branch, the platforms at Bank
and London Bridge are "reversed". i.e., why do the running lines
swap sides during this stretch?

In most places the early deep level lines were built under the road.

True, but I have not seen it suggested that that was the issue here.
It doesn't seem plausible to me. The line left the original terminus
at King William Street heading west and curving south under what is
now Arthur Street; this section *was* constrained by the width of the
road and had one tube under the other. They spread apart to the usual
side-by-side arrangement, but reversed, where they emerged from the
street to pass under the river. So there doesn't seem to be anything
in the layout there that would make the reversed arrangement more
attractive.

And the other end of the reversed section was where it is now, between
Elephant & Castle and Borough; the line runs under Newington Causeway,
which I've never visited at street level, but on the map it looks
near enough to straight, so there seems nothing to make a crossing-
over desirable there also.

As to the actual reason that the lines cross over, I don't know it.
My books are silent on the topic except for one, "Rails Through the
Clay", where there is only a vague statement that it was "related to
the original intention to use cable haulage". The original line was
authorized in two sections, north and south of Elephant & Castle, and
these were to have been built with separate cables (running faster on
the straighter south half) until the decision to use electric trains.
Maybe the crossed-over layout would have made it easier to lead the
cables into the tunnels (in the area where they cross over, rather
than at E&C station itself) somehow; but that's only a guess.
--
Mark Brader "Well, it's not in MY interest -- and I represent
Toronto the public, so it's not in the public interest!"
-- Jim Hacker, "Yes, Minister" (Lynn & Jay)

My text in this article is in the public domain.