How to replace old style cotter-pin bottom bracket with a modern one?
I did this just last year. It's not a hard upgrade. You'll need to replace the bottom bracket and crankset.  With a bike of that age, depending on the make, it might be difficult to find a modern bottom bracket that can fit your bike.  Take it to the local bike store to determine the shell width, or measure with your current one. This requires being pretty precise, as there are various standards and they all vary by only a couple of millimeters. 

Get a crank set that matches the number chainrings you want (looks like only 1 from the pictures), and is compatible with the new bottom bracket and you are set.  Square taper is a good choice for upgrading old bikes as it's cheap and ubiquitous, and works great.  A bottom bracket tool for putting the new cups in is quite inexpensive, as is tool for tightening the crank arms.  You might already have something sufficient for this as it isn't a special tool, usually just a larger allen key, although you'll need a lot of torque, so one that fits on a socket wrench is advised.  

If you are unsure about doing it yourself, your local bike shop can do it in 15-30 minutes, and shouldn't charge too much.

My bike wasn't as old as yours, but I'm pretty sure English bikes from the 1950s still used either standard 68mm British bottom brackets, or in the case of a Raleigh, a slightly different proprietary size. 68 mm is still the standard so you're in luck if you have that size.  I'm pretty sure the Raleigh size is still available as they were quite a popular make of bike.