Yes, Sometimes

My question is, if you do acquire this manuscript, how will you go about choosing the illustrator? Does a particular illustrator immediately pop into your head,

Yes, sometimes.

or do you take the marketing advantage of teaming a new writer with an established illustrator into account first?

Yes, sometimes.

If neither of these is true, do you go through postcards you've received, talk to your Art Director for suggestions, or wander online portfolios (which ones?) or illustration annuals (which ones?) to find the illustrator?

Yes, sometimes.
(Ones the design department suggests.)

How much time do you generally spend on this process?

As long as it takes. Sometimes that's a week or two (ie, no time at all: knowing what illustrator I want: illustrator agreeing to do it), other times it's a process of a couple months as design and I find time to meet a couple times to discuss ideas and narrow down who they'd like to work with and who I think will give the project what it needs.
The moral: What's right for one project is not the same as what's right for another.