If you can improvise some sort of press to push the cotterpins out, it is probably better but hammering will work if done correctly.
Remove the nut, take off the washer and then put the nut on again just enough to cover the end of the threads and then support the crank and hit it (very) hard with a hammer. This is key - one or two good hard whacks and the shock should free it. If you just sit tapping at it, you'll end up riveting it in position. Once it moves, remove the nut and drive it out completely. It should be possible to re-use them if the threads are good. If it's worn to a notch where it meets the crank, just file that flat but make sure both are filed the same amount or the cranks won't be in line.
All British bikes use a 3/8" cotterpin. 9mm are also available but that is for French or German bikes.
And when re-fitting them, it is important to press or hammer them into position. The nut's only purpose is to keep it in place and the threads are not strong enough to pull it into place by tightening.
I used to have one of these exact bikes. It was the very first bike that I reconditioned and it is the bike that got me into vintage bicycles! Now I run a full time business selling vintage bike parts and I'm also a collector, all thanks to this bike!