My DD was allergic to cow's milk (grew out of it aged 6) - I bf until she was 1, but used soya formula in food, and as a drink from about 6months.
You can get hypoallergenic formula, but we were advised that it tastes rather different, so to try the soya milk first - tiny bit on skin, if no reaction big bit on skin, then tiny bit on lip, bit more on lip, small amount in mouth etc etc. We got the soya formula on prescription, but it was available off the shelf too (Boots, and probably other similar places).
Ask for another paediatric appointment. Complain if necessary that you have been given incorrect advice (no replacement formula!?) - when I was told by GP DD probably didn't have an allergy (so I should just try some more milk and see what happened!!) I went back with "I'm probably just being a fussy mother, but I'd like an allergy test, so that I know for sure, please". Who cares if they think you're being awkward - you're doing it for your son. You should also be given an appointment with a dietician - ours was fantastic. (I found HVs no help at all.)
I cut dairy out of my diet, seemed to make her less sicky.
We were advised not to try egg until DD was 1 (reacted - still allergic to that), and to keep off nuts until 5 (then 7). She's fine with nuts, and I don't know what the current advice is, it may be different.
Good luck with everything. For the future - there are loads of dairy-free foods that you can get - it is awkward, but there are worse things.