My advice would be to check the bag through if you have to check it in LHR anyway. It will give you more open options to decide whether to exit to UK or not. UA will happily check the bag with whatever carrier.

Transfer to T4 is not that awful, airside transfer is the same as to T2 while for landside you indeed need to take either HE or tube. The question is more about which security line is bigger and without status or C ticket this may be a BIG question in the morning. Of course if you go landside you will have access to UA arrivals lounge.