alfie my thing is that I can't cope with being sick myself - if it's someone else I'm ok unless it's something I think I can catch.
So milky sicks etc weren't too bad, and they kind of ease you into it tbh! Food sick not so good, and DH works away a lot. We've had the odd time when they've just randomly been sick, like when DD had egg (turns out she's allergic!) and if DH isn't there I do struggle, but you just have to do it. I can't cuddle them when they're all sicky which does make me feel bad 
but we've been SO lucky in that they've never (touchwood!) had a bug or anything like that yet, just colds etc. I don't know what I'll do if/when they do.
Making a plan helps me - I have bleach spray for the bathroom, and towels from bedroom to bathroom if I suspect anything, then it can all go under the hose then in the wash, and bleach everything.
Honestly though, since having children I have improved so much. All the milky sicks etc have helped, and also the fact that you just can't really get away from it. And I'm super conscious of not wanting to pass my fear on to them.
Ooops that turned into a bit of an essay!! 