From a (nostly) Australian perspective:
Do you want Masters by research, or by coursework? I assume by 2A you mean 2nd class 1st division (normally this would be called a 2-1).
Most research Masters degrees (in Australia) are 2 years, and you typically need a 2-2 (or a Desmond... geddit?) for entry (a credit average equivalent), a 2-1 is only needed for the vague hope of an APA scholarship (in practice a 1st is needed for that).
Coursework Masters are typically aimed at professionals to enhance CVs and skill-sets, so they're normally not the route to a PhD (especially in maths), and are usually 1 year.
My experience of the UK system is that it's largely similar to the Australian, though I have spent less time in it, a 2nd class degree is all that's required for entry into a Masters (so 2-2 would be sufficient).
The US system is somewhat different though (but warning, my experience with the US system is from the outside, so take this with a grain of salt). For those looking at a PhD, you typically join a graduate programme, which has 2 years of coursework followed by 3 years of research largely as a single continuous (or at least contiguous) programme. Often the 2 years coursework will gain you a Masters degree, but this is a "Masters in passing", and is not strictly necessary for progression (this set up is also why the entry exams might turn up at the start in the US).
So after all that, what you want to look for is "Masters by research" programmes, normally each department at the university will have some information about it on their website under something like "Postgraduate Study", or similar (perhaps "graduate school" in the US, but read carefully). These are the ones aimed at PhD preparation, though in Australia and the UK, there is typically no or minimal coursework, so if you need to catch up, it's up to you and your supervisor to ensure that this happens.
The less pleasant part is that if you only have a 3rd class degree, then you're going to have a tougher time of it. In this case you may want to talk to academics at the university you're interested in about what you can do to demonstrate appropriate ability. Another option is to bite the bullet and complete a 1 year masters course to attempt to demonstrate that you have the skills and knowledge required. A really extreme suggestion would be to complete another undergraduate degree specifically in mathematics (many universities offer a specific BMath or similar) to close the gap.
Finally, if you're in a position to go for a Masters by research, take some care picking a university that offers research in an area you're interested in. Think about what areas of maths you like, see where the people who do that are and talk to them. The vast majority of academics in Aus./UK/US are keen an interested to talk to people who want to get involved in their area and they tpyically don't stand on ceremony, so you don't have to be too worried about formalities (but be polite of course!).