These were played more by girls than boys, and I think their disappearance has something to do with the availability of real sports for girls now. There weren't many outlets then. Swimming - not much good on the playground or most neighborhoods; cheerleading - much less athletic then than now; gymnastics, ditto; riding bicycles or roller skating. Organised sports were bigger in the private schools, likely because they didn't have neighborhoods to go home to. Skiing for the rich girls, skating more middle-class. Athletic girls were often directed into dance.
Even more important, "Ten O'Leary Postman" can't much compete with the fascination of an electronic device.
I was surprised how many of these British games and rhymes had American equivalents
The song Rubber Dolly was originally a ragtime number that got adapted for bluegrass, then done by Helen O'Connell with Jimmy Dorsey ("Is it the feller or the dolly, or the dolly or the feller?"), eventually becoming a playground rhyme done by a girl group in the 60s that keeps popping up over and over.
My wife did not recognise it. At her schools in Massachusetts, as at mine in NH, "Oh Little Playmate" and "Spanish Lady" were the thing.