Fennel, The UK and the US have always had very different views on fundamentals. The NHS a case in point, they see "socialism " as the worst thing ever, and the NHS a prime exmple of socialism at its worst. Most of us see the NHS as the best of British.
Yes, they joined WW2 after Pearl Harbour but we have to be thankful for that late entry because our backs were well and truly against the wall. Thousands of Americans gave their lives to save Europe.
As for your question about how American republicans would regard Momentum, I'm not sure what you're asking. I'm a left leaner, life long Labour voter and I'm unimpressed by Momentum. I feel they're all to similar to Militant who come close to destroying the Labour Party in the 1980's.
The 'special relationship' means much less to me that the EU. I see myself as a European. I trust our European neighbours more than I trust the American government. I mistrust Trump more than any previous President in my life time, and that's saying something.