Last minute theatre tickets are on offer all over London and it's possible to get some very good deals on the day of the show from various ticket booths, online or even the theatre itself.
If you have left it too late and have gone to buy tickets only to be told the show you want to see is fully booked, then it's a good idea to go to the theatre in person on the day of the show as most theatres in London's West End offer day seats and last minute theatre tickets to be sold only on the day of the performance. These seats are often a fraction of the cost and usually well situated in the theatre. You may want to check with the theatre at what time to arrive as some sold out shows have people lining up for these tickets from the early hours of the morning.
Alternatively, as a last resort in a time of desperation to see their favourite actor/singer/dancer on stage in a completely sold out show, people turn to the ticket scalpers or ticket touts as they are known in the UK. These people work illegally outside the venues/events and show up with unsold tickets from brokers offices. There are many full-time touts who are regulars at particular venues and even have a league of loyal buyers. These people feel safer buying from a familiar face and expect that they are less likely to be ripped off. You can never be too careful though as there are plenty of scam artists who will sell fake tickets to unsuspecting buyers or even sell them tickets that have already been scanned entry into the venue only allowing one entry. It's probably best to give the ticket tout a miss and just wait for a future opportunity.
Before giving any of the above mentioned options a single thought, you should first get online. There are so many online agencies selling last minute theatre tickets for London shows and often, if you are online in the morning hours there is a good chance you will snap up some of the last legitimate tickets for the show that night. Just make sure the company you are purchasing from is accredited by S.T.A.R (which is the Society of Ticket Agents and Retailers) and you have nothing to worry about.