Cambridge do not take late applications and if wanting to consider it as an option, you should have applied by their October deadline. LSE and UCL require you to take the LNAT by December 31. KCL do not consider late applications until after all on-time applications have been processed, meaning theres a low chance you'll be accepted due to the competition.
If you wanted to go to the top UK unis then really it is a process you need to be doing way ahead of time.