In my personal experience as a native U.S. speaker, all three are common, natural, and frequently used, and I can't think of any distinctions of meaning or regional usage. I think my wife and I would most commonly say simply "gas the car" (no "up"). "I'm going to get groceries." "Oh, do you think you could gas the car on the way back, we're down to a quarter of a tank." These all seem like the same pattern: water the grass, feed the dog, gas the car. There is also to "top up" or "top off" (interchangeable, no difference) the tank, which means to make sure it is completely full, particularly when it is fairly full to begin with. "We've got three-quarters of a tank but I think I'll stop and top it off."

And in American English, we would most likely say "fill up the tank"

Well, we call it 'petrol' in the UK and not 'gas' so no one says 'gas up' the car. I know 'gas' is just a shortening of 'gasoline' but it seems counterintuitive to me to call something that is liquid 'gas.' As Su.Ki said, you 'get' or 'buy' petrol from the petrol station. Alternatively you might be less specific and just say you need to buy or get some 'fuel.' 'Fill up the tank' again is not likely to be heard for the simple reason that modern petrol stations don't have parking attendants in the UK, you fill up the car yourself so this phrase doesn't make much sense. As for 'tank up' again, it is not used. The only time you might hear this phrasal verb used is in the context of getting drunk (in slang) i.e if someone is tanked up then it means they are very drunk.

Not at all. A British English speaker would not say 'tank up' or 'gas up'. In British English, the word 'gas' is used in its literal sense, and refers to something which is chemically a 'gas' - like methane, propane and so on. The fuel which we put in a car is a liquid, so it seems illogical to the British English speaker's mind to call it 'gas' when it clearly isn't a gas. We might say 'fill up the tank' but it's not especially common. It doesn't sound obviously American in the way that the other two phrases do, but I don't think that we often say this. In the days when you had to ask a garage attendant to operate the pump, you might have said 'fill it up'. These days, people would probably just say 'buy' or 'get' 'petrol'. For example, I might say 'I'm just going to call in at the petrol station. I'll check the oil and tyres, and get petrol'. In general, American and British English are not nearly as different as learners imagine they are. However, when it comes to terms relating to car and road use, there are quite a lot of differences.

I am a British English speaker and have never heard the expressions 'tank up' or 'gas up'. We don't tend to say 'fill up the tank', so much as 'fill up the car'.