I am currently purusing a PhD degree in some second-tier university in US. I do not think you should waste any time in getting a master degree because there is no substantial difference between courses in two levels (some Cambridge Part III courses are definitely research level). The difference is during the first 1-2 years of PhD you decide your field and your advisor, in 3-4 year you work on your theis problem to graduate. While in a master program you are often not required to finish a thesis, and often has no formal advisor, usually only required to finish a certain number of courses.
However, to be realistic if you want to increase your chances of getting into a better PhD program and be better prepared to do research, then a one year master program is not a bad choice. At least it is better than going to a bad PhD program and work in relative isolation. Most PhD students learn material from seminars and books, lecture notes, etc, and often the advisor do not have time to guide students in every step during their research. Therefore it is necessary to know some basics in your field before you started research, and a good master program with reasonable difficulty courses might be helpful. Maybe you can try to write a paper in your field(may not be publishable, but still good exercise)? I would guess you will not have trouble to get a decent grade, references, etc after your master program, and you may get into a better PhD program afterwards.
Also an extra year may give you some elbowroom to decide what you are really interested; I came to my university without realizing there is no professor specializing in my field of interest, and I either need to shift focus or transfer out. So if you are applying for PhD, you should 'know' if your target school's professors' research is compatible with your personal taste. And if you are undecided(a lot of people are only interested in some abstract idea or mathematical structure), then imagine yourself is already a graduate student - who you want to be? what you are interested? with whom you want to work with? You may just apply and decide when the offer comes and school year begins, but I think it make sense to think about this when you started the application process.
I did not notice that you are planning to learn differential geometry. If yes, then you should check with the top geometry PhD schools in US, UK and France (maybe Russia as well) to see what their programs are like. Differential geometry does not require a lot of foundational reading like algebraic geometry, but it relies heavily on PDE. So you may need to study some PDE seriously. Most of what I wrote earlier still applies to this case (that is, a good master program preparation helps your chance to get into a top PhD program). But if you are convinced to study differential geometry, and believed you could not get a decent research experience in a typical master program, then you should apply for PhD.