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.