The origins of the phrase lie in the verb tinkle, which developed a transitive and causative use quite early: ‘to make something tinkle’. OED has some excellent examples from the 16th and early 17th centuries of instruments tickling: ‘Many drums were beaten and basons tinckled about them’, for example. And then in 1817 we find this lovely specimen from a familiar name: ‘She was very fond of tinkling the keys of the old forlorn spinnet’. Jane Austen. Northanger Abbey.
At around the same time, the piano keys were beginning to be called ivory. John Keats is the first citation in the OED: ‘She plays the Music without one sensation but the feel of the ivory at her fingers’. A plural usage probably came soon after, but the first OED citation is not until 1918, ‘the black and white ivories’.
Meanwhile, a transitive use of tickle was also developing – at more or less the same time as tinkle - meaning ‘to touch an instrument lightly, especially one with strings. Thomas Nashe is the first recorded user here: ‘to tickle a citterne’ in 1589. The usage continues - strings are tickled, a guitar is tickled - until we get to the early 20th century, when we find ‘tickling the typewriter keys’ (1926) and then (1930) ‘tickle the ivories’. The Times in 1962 has an interesting comment: ‘Ivory-tickling’ has become an outmoded and faintly derogatory description of piano-playing.’
So, tinkling came first, and tickling later. But there’s no suggestion of any transatlantic difference in the citations. On the contrary, both usages have solid histories in the UK, and I suspect the tickle one has had a great deal of usage in Cockney speech. My feeling is that this is no more than a parallel development where the phonaesthetic similarity – just a little bit of nasality before the /k/ - has made the two verbs seem interchangeable. But tickle the ivories is about three times as common as tinkle the ivories. Probably the ‘faintly derogatory’ sense of tickle has made it a less palatable expression over time.