The vet telling you he'll get ill if he only eat dry food is not very helpful, and neither is it true.
If he sin't eating after three days, he won't eat it and as he is a young puppy, I wouldn't attempt to leave it any longer. My eldest was weaned onto Iams (made by Eukanuba) by her breeder, and she was never really keen on it. I think a lot of dogs aren't really. You coul dtry putting a tiny bit of tinned sardines in tomato sauce with some warm wateront he food. Mine go MAD for sardines.
If you want to try sticking to a dry food, then try James Wellbeloved Puppy. It is not the best food, but it is easily available and the best of what you'd get in Pets At Home or pet shops, IMO. They also do puches of complete puppy food that may encourage him to eat. I have weaned pups onto James Wellbeloved and just put a desert spoonful of the pouch and some warm water to make a change for the pups.
If that fails, then Natures Menu do a wet complete food, also in Pets At Home which is fairly decent and certainly won't make your pup ill.
Brands like pedigree, bakers, Wagg are best avoided IMO, as they have a low meat content and contain additives.
Do try not to chop and change too much though, as it will upset your pups tummy.