But given the aftermath of the War of 5 Kings and the shock and awe of the dragons wouldn’t a lot of people turn and join Dany? Keeping Highgarden from burning and their own necks might be enough to turn over Tommen.

That is the most likely scenario, but the original question was whether Dany and Dorne could take Westeros militarily, not whether Westeros would choose to make a fight of it. 

And I highly doubt Dany goes up against Tommen. The vision in HOTU wasn’t of a mummer’s stag, after all.