Hi,
My 6 months old daughter has gone through the same situation.
When a baby has severe diarrhea, vomiting, etc... the digestive system is constantly under a lot of stress. The diarrhea washes away the ennzymes that allow them to digest (break down lactose) and also impact their capability to digest all other food. The digestive systems needs a break, a rest... most babies after months and in some cases years are able to go back to dairy products and that is due to the digestive system rebuilding those enzymes.
Your doctor was right to prescribe Neocate.
Although it may seem your baby is reacting badly to it, Neocate is the most elemental formula you can get, it is fully-digested, therefore your babies body will absorb all the nutrients without needing to digest it. This will give the digesting system a break and time to regenerate the enzymes that were lost.
If your baby is vomiting Neocate, like mine does... try feeding more often small quantities, also stopping and burping quiet often as it tends to build up a lof of gas (this is all due to their digestive system still being so fragil). Do not move to NUTRAMIGEN, that is a partially-digested milk. Even less to Cow&Gate, Soya, or SMA LF formula... these are all non-digested formulas... and would demand too much from your babies digestive system. When I switched to SMA LF... Wow, big mistake she went back into diarrhea and this time acid poos, that gave her open wounds in her bum and a very painful rash, she would cry and scream to pass every single poo (at times only 2 minutes apart). Keep with Neocate for now.
For the constipation, we give our little one a 100mls of pure water at around 6am, and it seems to work... she would poo sometime between 7am and 10am. Also do not use the full strentgh, maybe one scoop less (for the same 210mls bottle), this will also help with the constipation. For the taste that is terrible, we are boiling the watter and when colling it we add a vanilla pod, we then remove the pod before the feeding. It improves a lot the taste.
Wish you the best of luck! I know exactly what you are going through!