It's easier to find the probability that there will be no game between British teams, then subtract that probability from 1 to find the probability that there will be at least one game between Brits.
There are possible ways to pair the teams into games without any restrictions (a multinomial coefficient). All of these arrangements are equally likely.
How many of the pairings result in no British-British pair? Well, there are 4 possible ways to match the first British team with a non-Brit. Once this selection is made, there are 3 remaining non-Brit teams to pair with the second British team. Then there are 2 possible pairings for the third British team, and finally only 1 choice for the fourth British team. So there are ways to pair the teams with no British-British game.
Therefore the probability of no British-British game is p = 24 / 2520 = 0.009524, and the probability of at least one British-British game is 1 - p = 0.9905, approximately.