They are all more or less the same, i.e. they all mean "it annoys/irritates me a lot" or "it gets on my nerves" and are often prefixed by "really" for emphasis. From a London/SE England perspective, I would say "winds me up" is the most common (and least slangy), with "hacks me off" the second most common. I find "gets my goat" a bit old-fashioned but maybe that's just my impression - it still gets used quite frequently. I have heard "grinds my gears" too but I haven't heard that as often as the other phrases. There are also similar, more vulgar phrases and there are no doubt other ways of saying this that are specific to certain parts of the country too.

"Gets my goat" is something that my mother might say. It seems a bit antiquated. "Hacks me off" is something that I have not heard for a long time. "Winds me up" is something that I have said from time to time. "Grinds my gears" is pretty common. Honestly, what I usually say is that something really "chafes my ass" (a bit more rude, but it gets the point across). And yes, all the above are synonymous. I don't know which are more common in the UK.

Get's my goat is quaint. You can say it if you want to be overly cute about something. Winds me up, yes. Never heard "hacks me off". Wonder if it's a British vs. US thing. Grinds my gears is probably getting more common: https://www.youtube.com/watch?v=Q685Ko2DHDs

Thanks for that question Sandra! I learnt a lot today!