“Give me a call” or “ring me.” The phrase is sometimes shortened to “give me a tinkle.”

“Tinkle” refers to a phone’s ring, while “blower” is slang or telephone and refers to the device that predated phones on Naval ships. Sailors would blow down a pipe to their recipient, where a whistle at the end of the pipe would sound to spark attention.

“Give me a tinkle on the blower.”

“Give me a call” or “ring me.” The phrase is sometimes shortened to “give me a tinkle.”

but it is one of those phrases that is so cliched that you wouldn’t hear anyone in the UK actually saying it seriously to someone. It’s the sort of phrase that Americans pretending to do a British accent might say.