The exact equivalent, I would say, of the U.S. get along with; used by Dickens in Bleak House: “They get on together delightfully.”
It’s an example of the sort of Britishism that was popular with mid-(twentieth-)century intellectual Americans, such as
- Novelist Diane Johnson: “Depending on how you get on with her–she is the most important figure in the book (as we will see)–you will learn about how Constance Philippa escaped on her wedding night.” (1982)
- New York Times columnist Flora Lewis: “Nothing different is to be expected from Mr. Qaddafi. If anything is surprising, it is that countries that have been subjected to his cruel wiles still imagine they can get on with him, do business with him, appease him.” (1986)
- and editor Robert Giroux: “I had to get on with him and I made sure that I did. ( 2000)
It is now emerging as a NOOB, for example in a December 4, 2011, lifehack.org post by Paul Sloane, which asked: “What should you do if you really cannot get on with your boss at work?”
This is, of course, distinct from the verb phrase to get on with it, which goes to the very heart of the British character, which was used by Lisa Simpson in an episode a couple of weeks ago, and which will be the subject of a future post.