Applying to US Phd Programs while doing a UK Master's I recently received a conditional offer from UCL in MSc Advanced Neuroimaging. I'll be completing this program through distance learning (meaning that I won't have to physically be present in the UK). 

I was thinking about applying to PhD programs in the US this fall with the hopes of starting in the fall of 2020. However, while I was looking at the course requirements for the MSc program, I noticed that my thesis would require a final presentation in September 2020 and results will be notified in October 2020. 

Will this overlap (starting a PhD program in September 2020 while finishing my master's thesis) cause a problem? Will I have to wait to apply in the fall of 2020? I would really like to apply this cycle so that I won't have to waster another year.
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.