Britain in the 1890s KNEW that it was going to be replaced by the USA, that's in all the fiction and non-fiction books from the UK at that time. So they determined to make the USA a strong ally and trading partner and to make the USA "follow the British cultural mores" -- and they succeeded, just as they did to some extent with India, Singapore, etc.
I'm not so sure about that.
That period was the sweet spot of the Victorian era and jingoism. It's not much different than some of the crazy Freepers and other naive souls of today.
Their view was that "the sun never sets on the British Empire."
Except that it did! Just like so many before.