Anyway, I emailed my MP about this in February, and I got a reply from his PA a month later:
Dear Mr Kirk
Do you still wish to climb Big Ben? If so, could I please have the following information from you. Full name and address, date of birth and place of birth including town and country. This is because a security check is done on everybody who wishes to go up the tower. Also, you must be a British Citizen and be in good health. Are there any dates you definitely cannot do – if not, it will be much easier to find a tour time for you. Is it just for yourself?
So, before I reply to that, would anyone else like to come with me? If so, please send me the relevant info (email is probably better than a blog comment) and we'll go from there. If nobody replies by Thursday evening (14th) then I'll send my reply on Friday morning (15th).
According to the website, tours are now fully booked until the end of June, so they'll start taking July bookings on 1st April. The times are 09:15, 11:15, and 14:15, Monday-Friday. No children under 11 are allowed. My MP's original auto-reply said: "Please note that there is a strict rule within the House of Commons that MPs may only act on behalf of their own constituents." I don't know what happens if a group of people from different constituencies want to go together; hopefully it can all be booked through one person, rather than having to contact all the relevant MPs.