It is harder to socialise a puppy once it is past the socialisation window, but it's not impossible.
The fact that he is very nervous isn't necessarily a sign that he hasn't been properly socialised, but I think in your position I would want to clarify what socialisation had occurred before he came to you, so you can build on it. 16 weeks is old for a pup to move to a new home, but reputable breeders do sometimes have older pups, so the vet shouldn't assume the worst just from that - a breeder may have been let down by a potential buyer, or they may have been planning to keep a pup for themselves as a show prospect - in that case it's not unheard of for a breeder to keep two for a few weeks to see which is showing the best potential and then sell the other - these pups would be well socialised and cared for during this period.
We have a puppy at our puppy classes at the minute who is very nervous - last week he couldn't cope with doing anything in class, barked at every movement of anyone else, and we had to leave the hall twice to let him calm down, even with constant drip feeding of treats. This week, while he was still nervous and needed a lot of treat distractions, he was actually able to work on the exercises, didn't need to leave the hall once and, while there was some barking, he was quickly distracted back to focus on his owner.
Try and approach everything with lots of treats - I'd be tempted not to feed any meals in a bowl at the minute, but use his kibble as treats - if you can give him a treat whenever he sees or is faced with something he is nervous of, but before he starts reacting to it, you can actually change his emotional response to things that frighten him. Your OH can hand feed him (assuming Bear will cope with this - if not you can hand feed him while your OH is nearby but far enough away for Bear to handle it).
Also, your pup could be going through the secondary fear stage (it's a little early, but not unrealistically so) which could explain why he is now scared of things he was previously ok with. This is an entirely normal, if little mentioned, developmental stage...
I would also agree with Holly Berry, in saying that T-Touch could be very helpful...