The same reason we got involved in WWI and WWII. Because it's Mother England. It's the reason ABCA Armies is a much closer relationship than NATO or any other alliance we've got. Blood is thicker than water. Obviously the blood aspect is a little more dilute than it used to be, but we are ultimately Albion's seed - if not by blood, then by language, law and custom, among other things. The Brits paid for their own deployment in Afghanistan and Iraq, and they paid back all of the Lend Lease Aid we lent to them, in contrast to most of Europe, whose debts had to be forgiven. To the extent we have any friends on this planet, the Brits, the Canadians, the Australians and the New Zealanders are it. And it's because of a common heritage.
Apart from ties of blood and sentiment, it makes sense to court England, if only to pry it away from the European Union. The country is in the midst of a bitter divorce. Why not invite it to an all-expenses paid hunting trip (i.e. provide submarine basing gratis), if only to take its mind off its troubles?
Rotherham is their new heritage