Is there any major problem in applying to PhD program in US after getting BSc and MS in UK?
Let me clear up an important distinction between graduate school in the US and the UK:


In Europe, the master's degree is generally a pre-requisite for admission to a PhD program.
In the US, the master's degree is not required for admission to a PhD program, and you normally apply to the PhD program with only a bachelor's degree. You can obtain a master's degree as part of the PhD program, but this is by no means a universal requirement. It varies between universities and even departments within a university.
In the UK, options are somewhat ambiguous. You can apply for a master's and PhD program at the same time, while others may require the PhD in advance.


Therefore, if you are admitted to an American PhD program, you will take courses in graduate school, because that is the equivalent of the master's phase in the UK.

Other than that, there's no inherent advantage of applying or attending bachelor's or master's programs in the US or the UK.