How feasible would it be to transition from a Master in Finance to a phd in Pure Maths?
Bottom line is that it will be very hard since you lack a bachelors in math (or even the math training of a physicist).

My advice would be instead to go work for a while.  It seems like we get a lot of degree change requests which are prompted from not wanting to leave the nest.

If you don't follow the go work advice, I would at least look for something a little closer to your training.  Something in quantitative finance or econ or at least applied math or statistics.  Not pure math.