Hmm... There are plenty of places in London where you might be able to get some 'Work Experience' (from a few nights to a few weeks) but it will probably be much harder to find a place that could offer you a formal 'Internship' scheme, just because they don't really exist in the UK (not for English anyway).
Trying places like the RSC (not in London though), NT, Old Vic, Donmar etc. will probably be your best bet as they are producing houses. From your perspective an Internship on a West-End show wouldn't provide you with anywhere near the same scope or experience as a placement at a producing house would just because in the West End you do exactly the same job night after night!
If you were prepared to look around outside of London (Like Bristol, Manchester, Birmingham, Cardiff etc.) you may have even more luck.
Let us know how you get on and where you end up!
Good Luck!
Sam x