Bush's power over Blair

That is to misunderstand the dynamic between the two. They had something of a personal rapport, which whichever way we wish to look at it, May & Trump do not.

Bush did not hold the whip hand over Blair. In many ways it was the inverse, with Tony wielding a great deal of influence over the man. They both wanted Saddam out. It was down to conviction, albeit an ill fated one, rather than a power dynamic.

how much power Trump has over May, especially given the trade deal situation.

This is a very good question, one that does not necessarily correlate with Iraq as I set out above. Firstly do we believe she's capable of such a terribly corrupt thing? The DUP Deal suggests it's a possibility. But the Macron factor suggests it may not be. In fact Trump may have even been reluctant, we don't know.