In the cards or on the cards?

The Cambridge Dictionary of American Idioms says both forms (in and on) are used, and that the origin is in Tarot:

in the cards also on the cards

based on the use of tarot cards (= a set of cards with pictures representing different parts of life) that are believed to be able to show what will happen in the future

But, despite the mystical origins, in both the US and UK, the idiom simply means "very likely to happen":

very likely to happen ♦ I think winning the World Series this year is definitely in the cards for Boston. ♦ Some reports suggest that a tax cut is still on the cards.

Note the the very American bent to the first example (not to mention this is the "Cambridge Dictionary of American Idioms").

On a personal note, I've never heard "on the cards" used here (East Coast of the US), and somehow it sounds illogical to me; I understand "in the cards" to mean "in the reading of the cards", or "in the forecast of the cards", that is, the cards somehow contain the prediction in a metaphorical, abstract sense. A commentor (Jon Jay Obermark) also mentioned that in the American Midwest, the phrase is understood as relating to playing cards (in particular poker), and "in the cards" means "[inevitably] due to the arrangement of the deck prior to it being dealt", which also calls for the "in" (because the deck already contains the outcome).

By contrast, "on the cards" strikes my ear as firmly locative (the only thing I would expect to be on the cards is a stack of poker chips holding them down, or maybe some mustard).

In the cards (or on the cards)

• Likely or certain to happen, as in I don't think Jim will win-it's just not in the cards. This term, originally put as on the cards, alludes to the cards used in fortune-telling. [Early 1800s]

• very likely to happen I think winning the World Series this year is definitely in the cards for Boston. Some reports suggest that a tax cut is still on the cards.

• Etymology: based on the use of tarot cards (a set of cards with pictures representing different parts of life) that are believed to be able to show what will happen in the future

Source: Cambridge Dictionary of American Idioms Copyright and (TFD)

Be on the cards: is mainly a BrE expression :

• UK ( US be in the cards)
• to be likely to happen: "So you think they'll get married next year?" "I think it's on the cards."

(dictionary.cambridge.org)

To sum up, in the cards is the more AmE common expression, while on the cards, referring to something likely to happen, is the more common expression in BrE.

I never encounterd ‘in the cards’ in the UK or Ireland. When I first heard it in American films I thought it was a mistake.

Here in Canada, as in the US, it's "in the cards", which to me suggests something interpreted from a tarot reading; an understanding arising from the combination of signs; akin to "in the stars". I get "on the cards", but to me, that sounds like what you'd get if someone sneezed during a poker game.