MS in Pure Maths to PhD in Statistics
I think the main hurdle will not be the knowledge about statistics you'll bring. (As a trained mathematician you should be able to pick up new math quick enough.) The main hurdle could be to embrace a new kind of doing science and research.

In more applied math people sometimes think differently about problems than in pure math. Roughly speaking, one is much more open to adapt assumptions and requirements for some problems such that they become more simple theoretically, more suited computationally or otherwise more convenient. Since one usually works with things that are related to the real world and hence, somehow uncertain and noisy, it is often not useful to stick to every assumption but use some freedom in modeling. I heard the saying "All models are wrong." Well, true but this should not make you hesitate to work with models.

I've seen people changing from pure math to applied math who never could adopt to these principles (but also cases where there were no problems). If you think that you won't have problems with issues like I described, then things will turn out fine, I think.