You could start on a half-dose while you're sick maybe if you want to get started with it... I don't think prozac usually makes people feel ill at all but maybe your idea is the safer bet. Prozac is usually capsules I think but you can dump half into yogurt or something and take the other half in the capsule if you are eager to get started on it.

Paxil makes me feel a little spacey when I first go on it but have taken a few others too and never felt anything I would describe as feeling sick or ill at all. Prozac is a really proven one with high success rate, but don't get discouraged if you have to try one or two others before you find one that really works well for you. And if after 4-5 weeks you feel notable improvement but just not enough make sure to talk to your doctor about potentially raising the dose (or trying another one if you don't think the current dosage of prozac is helping significantly at all).

Either way hope you get over the cold soon! I'm happy to hear that you're set to get started now on an SSRI and don't have to wait for a 2-month-out appointment or anything

Take care