I was initially looking into mechanical and so was going to apply to Southhampton, Bath, Nottingham and Warwick but now I am not sure if they offer good courses for any of the three engineerings above.
Especially for Southampton I had a friend who did Mechanical/Mechatronics there and he said it was great.
Usually robotics and mechatronics are done as part of an engineering course (particularly in the electronic and mechanical engineering courses) but nonetheless, I'd suggest having a look at Lancaster, King's College (for electronic engineering), UCL, Brunel and Surrey are some I consider really good and usually are rated highly in the rankings and on student satisfaction. Also Queen Mary, Manchester, Loughborough and Heriot-Watt are good options.
Of course, the one and only, Imperial College is probably the best option but I don't know where or what you're aiming for.
Hope this helps.