Basically it is possible to be an au pair for a short stay of only 3 months in London. However, the more flexible you are with the conditions of your stay, the simpler and faster it is to find a family. Keep the following points in mind:
1) Don't only look for host families in London. Great Britain is the classic host country and English is the number one global language. Many young people want to go to London. Think about whether you might not also be able to stay in another British city or in a smaller town.
If your main aim is to improve your English language skills, you could also think about an au pair stay in Ireland or one of the Scandinavian countries. English is also spoken there in many families.
2) Be flexible with the length of your stay and with the beginning and end date of your stay. Instead of entering a definite date in your search criteria, put in a wider time period.
Host families tend to be looking for an au pair for at least six months and most preferably for a full year. Therefore, think whether a stay of six months would be possible. That will increase your chances of finding a family enormously.