"Rink" is a Scots (!) word meaning a large open space for bowling, curling, or yes, skating.
"Tumble" (or if we're going to stick to Scots, "tummle") is doing somersaults, cartwheels, or perhaps dancing with wild abandon or just flailing about.

You can stick them together and get "rink-tummle' and it's your basic wild garden party or possibly flailing around in a place you're supposed to be bowling or curling or skating. (Sounds like an average hockey game.) You shorten "rink-tummle" to "rinktum," and voila!

...or not.