I have been mentioned.
First thing to do is contact the current marque owner, Martin Coopland. He can verify the build date for you. My email search function is just not cooperating or I’d give you his addy. He is active in a lot of ways and not too hard to find. If you want a fuller set of decals he has those too.
My 1950 was refitted by Oscar Wastyn in 1959 and had been spread to 120 when I got it. The slightly later spec has made it completely a rider. If you want to be fully period correct Iab is right, 114 would be the most likely OEM spec for gears. If you wanted fixed wheel either 110 or 114 would work and those are easier to find than you might think. Yes, the fork absorbs shock, no question about it. Check the alignment of that fork before you do anything else. Getting them straight is double the curves, double the work. And confusing. And worth it.
Just go get a Stronglight bracket for your Stronglight crank. Still common enough. The most frequent original crank on Bates of this period is Stronglight 3 pin steel. Either way is good. Old headsets are pretty available. If a basic Rewax headset is good enough, Andy at Yellow Jersey has supply for 3 lifetimes. English and period correct and cheap.
Measure the brake reach. Mine is really very long. Many Bates have this, not all. Mine has 1959 Weinmann 999-750 with the single spring. Pads are all the way completely to bottom. Much stronger brake than later Weinmann. If you wanted super correct it would be GB or other English side pull that wouldn’t much stop the bike. Only period correct that stops the bike would be Mafac, Mafac w/enough reach might be hard. Raid?
B.A.R. is a race bike. Used with tubulars. Old rims are still available, I think they are stronger than familiar 70s 80s parts. Hubs should be 40/32 hole rear/front. Hubs not hard to find. The 40 hole rim is the hard to find.
Gran Sport is just a wonderful derailleur. Prefer them to Record or Nuovo. That braze on shift fixture is completely wrong — but maybe not so bad. One of my other bikes is a 1958 Rivetts that has Campy 1010 drops and the Campy shift mounts. Very original, frame had never been built up. So Dennis The Slasher Beales could have put that on when it was first repainted. I wouldn’t take that off.
Older Brooks can be found. This kind of bike does look a lot better with a saddle that completes. Mine is a 1960 Swallow that had been given to me 1969 and had never found the bike that wanted it. Find a good one.
Getting every detail perfect is less important than getting the bike out on the road. I just would not ride mine with correct 1950 brakes and am never going to do linen wrap cable casing. Can’t care that the toeclips are 70s. Although if some old and workable clips came by that would be fun. If you make an effort to be respectful nobody cares if something or other is wrong.
If you succeed in getting that old bike rideable it is like nothing else. First thing you discover is the front wheel goes over the bump or hole so smooth you almost don’t even notice. Then the back wheel crashes hard into that hole because you didn’t think to lift your butt off the saddle. Maybe practice this with a basic wheel before sacrificing a 70 year old 40 hole sprint rim. After a while it teaches you to feel the tiny bump and float the back wheel. The fork is also the most precise steering fork I’ve encountered. Just two weeks ago today I was hit by a car while riding the Bates. The rubber side stayed down. I didn’t do all that much, it was like the bike took over and told me to sit back. When the bike does that much ya gotta like it.
One last would be that after you are done being wowed by the smoothness this is a normal and surprisingly modern bike. The head angle is steep and the fork rake is normal, weight distribution is normal. You ride this one like any other bike. No surprises. Easy from the start.