Well spotted - here's the version I know:

Billy, Billy, Henry, Steve
Henry, Dick, John, Henry Three.
Three Teds, then Richard Two;
Three more Henries - then what’s new?
Two more Teds, then Dick the Bad,
Two more Henries, Ted the Lad.
Mary, Bess and Scottish Jim,
Charles but they beheaded him!
One more Charles then Jamie Two;
Bill-and Mary, Anne - then who?
Four Georges, Billy, then Victoria -
These lists of names were sent to bore ya!
Ted the Seventh liked his fun,
Georgie was the quiet one.
Ted came when Mrs Simpson beckoned,
Then came George and Bess the Second.

Is there one for Presidents of the USA?