egg/milk allergy in baby- tell me all you know please

[unreasonable]

a while ago I noticed that ds2 (now 17 weeks) seemed to react if I had a lot of egg (he is ex bf). he wwas very unsettled and sick a lot.
this seems to be getting worse and I have cut egg out of my diet completely for around a week now.
My first question is:
how long do I need to cut it out before I ought to see an improvement in him (if it is egg causing the symptoms)?
If symptoms haven't disappeared then is it possible/likely to be milk protein intolerance or something similar?

He is now having the following symptoms which can be matched to some sort of intolerance/allergy:
clear vomit (read that it could be stomach acid)
being sick a lot in general, which he never really used to do
persistent nappy rash which has appeared in the last couple of weeks
slight exczema problem
spots on face

any advice?

oh and he arches his back a lot, which i have also heard can be a symptom

thank you for the reply (just realised I still had my name-change on from an AIBU thread lol)

do you think I should carry on just excluding egg for a couple more weeks?
then cut out other dairy if he is still showing signs?

is it worth me going to the GP and seeing if he can be tested for intolerances?

haha yes, I suspected you may say that about GP's! it wouldn'e be the first time I have heard about their apparent scepticism with all things food intolerence-y. especially when concerning bf babies.
I think I may go, although we haven't got to the point of blood or mucus in nappies- so perhaps they will just say he is a sicky baby.

I am a vegetarian, so quite careful about my diet already. though may get some supplements to take if I do cut out the dairy et al, to make sure we're both getting what we need.

thank you for taking the time to reply, it really is useful to hear someone else's experience of this