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Standard gear, to the death.

Round 1: Book versions
Round 2: TV versions

Who wins and why?

Jamie, in the show, at least.

Is this two handed Jaime?

A big disappointment we'll never see this showdown.

No clue about the books.
Jamie in Round 2.

Book Jaime wins easily.

The show versions are definitely closer as they nerfed Jaime quite a bit, but he should still win.

Book Jaime would stomp, being much more formidable relative to the show (his fight with Brienne was much closer despite being at even more of a handicap and book!Brienne if nothing else being physically larger and stronger than her show self), being quicker and even physically stronger than Drogo, and having far better equipment.

Show Jaime still wins, but it's closer due to Drogo actually having a fairly impressive feat and show Jaime being less impressive (he's physically weaker and not as overall impressive). He still has a significant equipment advantage however, which should make his victory ultimately inevitable.