Restoring BSA bike - replacing rear wheel and freewheel
Getting the freewheel off:

You need a special tool that fits in the notches in the threaded part of the freewheel, which you may or may not be able to find. In any case, the freewheel is likely worn and it will definitely need servicing so you are not going to want to re-use it. Just get a new one.

A note on wheels:

Older bikes often had rim diameters that are not the same as commonly available modern wheels, you'll need to be careful about rim sizes. A slightly different size may fit the frame but you have to make sure the brakes can be adjusted to reach the rims. 

Beware that the 'size' commonly given for a wheel is not the actual rim diameter. See this Sheldon Brown page for details: https://www.sheldonbrown.com/rim-sizing.html

You'll also want to be similarly careful about the frame dropout spacing and axle length. See this SB page: https://www.sheldonbrown.com/frame-spacing.html.

On retro-fitting a derailleur system:

Almost certainly not possible as your frame has too-narrow dropout spacing to take a wheel with a multi sprocket freewheel. Also, it will be expensive and a huge pain to do. better to just buy a cheap used geared bike.

On retro fitting an internally geared hub:

Technical possibility. I believe Shimano and Sturmey Archer make internally geared hubs for a range of frame dropout widths. You'll obviously have to consider whether the cost is worth it for you.