Slow down. Think about what you have there. Think before spending money. With handcut rear dropouts that have fender eyelets that is certainly an old frame. At first glance the fork does not match. But maybe it does. This is old and everything was variable if not custom when this frame was built.
What doesn't match about the fork is that it is very short rake and very stout. A bike to be used with fenders, much less for grass track, just doesn't need that fork. But we really don't know. The main frame (certainly very old) has some stout seatstays, where most old stuff would use something lighter and more comfortable. Half a dozen reasons why it might have been built with stout stays and forks and we don't know. Was this built for a pure track sprinter? Some things can be known. One is you aren't using fenders if brake reach is honestly 37mm. Another is this bike is going to ride very harsh. You can bash around on it on public roads and have a grand time, you'd be a masochist to plan long rides on it.
So check the brake drop again. This bike would have been made for what the Brits called 'sprints', that is sewup tires. Same size as 700C clinchers. Were you measuring with a 27x1-1/4 rim? The most popular brake for bikes like this was by far the Weinmann 500. Minimum reach 40mm. If you measured with 27x1-1/4 and measured right then the correct measure for sprints is 41mm. Clubman above made you a very generous offer and you would do well to accept it. It is one strong brake. The fixed wheel is itself a second brake. There is no crime in drilling the rear bridge and having a third brake.
On the other thread you mentioned a new production Campy crank and a Soma handlebar. Well, like Rudy says if it is all silver it will look OK. And the Brits themselves commit such acts on old frames all the time. But why? For the cost of a set of new Campag cranks and a new handlebar you could kit out the whole bike in vintage. A vintage bike rides like a vintage bike. A Frankenstein bike rides like a Frankenstein bike. You have one great vintage frame there.
One place you might want to spend real money is tires. Get the softest ride possible with that fork. You can use tubulars and be guaranteed a plush ride. Clinchers maybe try Compass EL or Vittoria Corsa G+ Isotech. If not using fenders measure carefully the biggest tire you can get under that fork crown.
From a great distance on a computer monitor that looks like an older repaint in real British stove enamel. I'd keep it but find some nicer stickers.