All 3 of the terms can be used interchangeably with me. They all have exactly the same meaning to me, and I believe it is so with most of the Native English speakers I know in the US.

Here's an example of US usage (and a true story). In my childhood home, there was a landing on the stairway between the first and second floor. A landing is a small, flat, level platform. In order to fit the stairway into a small space, it was U-shaped. A flight of stairs led from the first floor to the landing, then a second flight of stairs led from the landing to the second floor. You walked up the first flight of stairs, took a couple of steps across the landing, and walked up the second flight of stairs. Therefore, the "stairway" had "two flights of stairs" and a "landing." I've never thought about "stairway" and "staircase" before. I think we use both in the US and I don't think there is any difference in meaning.

OK: 'staircase' and 'flight of stairs' mean the same thing: stairs inside of a building, e.g. in a house. 'Staircase' is American, and 'flight of stairs' is said in England. But a 'stairway' is usually outside, leading down to a garden for example, or a fire-escape leading down a building, on the outside. But do remember that people will get these mixed up, and that these rules are not fixed. You may hear English people say 'staircase', and people may talk about an indoor 'stairway'.

I've said and heard "flight of stairs" an innumerable number of times here in the USA. It's as American as it gets. I have no idea what they say in England.