Hmm. Could be that this is labelled incorrectly.

I always thought that the footbridges that crossed through the platform
tunnels were a feature of the Hampstead Line, and not the C&SL.

To me, this view looks like the bridge is about cross one of what is now the
C+ branch platforms, and on the otherside there would be a staircase down in
between the two platforms.

BTN

I thought that when I first read it, but when I next went through
Euston, I had a look at the northbound City branch platform, and there
are two large square holes high up the wall at the back of the
platform, about 1/2 way along the platform. The holes each look like
they would be the other side of the photo.

It can't be a cross bridge on the Hampstead line - there aren't any
holes like that on the platforms at Euston for that branch. Besides,
in the photo, there isn't a hole on the opposite side of the platform
tunnel - so the bridge would have had to have come down from mid air
over the platform. If it was like that it would still be pretty
unusual - more like a metropolitan line thing like at Farringdon,
where the stairs are over the platform itself, rather than off to the
side.