Applying to US Phd Programs while doing a UK Master's
I'll speculate that most places wouldn't find it to be much of a problem, but it is up to the accepting institution, certainly. 

However, you will need to get accepted based on your current record without the MSc and will need to present sufficient evidence of success in the doctoral program. Since most US doctoral programs in the US don't require completion of a master's first it wouldn't seem to be an impediment to be doing both. But you will need to be sure that the required work on either of them doesn't impede the necessary work on the other. It would be sad to fail at either because of overload.