Hello Arthur & greetings from the USA !
The smaller diameter copper tube you ask about is usually 1/4" outer diameter. It is an unloader line. The "switch box" is actually a pressure activated switch. As the air pressure in the receiver tank drops, this switch closes and starts the compressor motor. When the required pressure in the receiver is reached, this switch opens.
At the same time, there is a small "arm" off the switch contact mechanism which opens and closes a very small "poppet" type of valve located at the base of the switch. This is the unloader valve and has perhaps a 1/16" or 3/32" diameter discharge orifice. When the compressor is not running, this poppet valve is opened. When the compressor starts there is a slight time lag, during which the orifice remains open- allowing an unloaded start for the motor. Similarly, on shut down when required pressure is reached, this poppet valve opens and allows an unloaded shut down.
I imagine you have a UK version of the Ingersoll Rand compressor since you refer to it as a "Model B". In the USA, this would be known as a "Type 30" (unless I-R has changed their model designations- type 30 has been around forever). In the USA, these pipe connections would be 1/8" NPT (National Pipe tapered thread). In England, I imagine you'd be looking at a British tapered pipe thread, but stranger things have happened.
My advice to you is to go to a pipe supply or even a good automotive supply shop. Ask for some compression type tube fittings having 1/8" tapered male pipe threads x 1/4" diameter copper tubing. These should be brass fittings and relatively inexpensive- rather than trying to match up to what I suspect are the remains of two compression adapters (an adapter goes from one system of pipe joining and pipe to another- in this case, pipe to tubing and vice versa).
If you want to verify the thread size of the tappings in the head of the compressor, try screwing in a 1/8" threaded pipe nipple or plug. The bottom of the pressure switch usually has the male portion of the compression fitting made integral with the switch rather than pipe threads. Unfortunately, with compression type fittings, when the connecting piece of tubing is removed, so are the mating parts of the compression fittings. These would be the hollow ferrule (oldtime British term was an "olive", I believe), and the gland nut. You may have to buy a couple of compression fittings and see what parts work on that pressure switch's compression fitting body.
Otherwise, you are likely looking at replacing the pressure switch if nothing matches up. I have the same style pressure switch on an ancient Worthington Compressor in my home garage/shop. Using a micrometer and screw pitch gauge, you can get the thread data on the male end of the compression fitting that will be projecting from the underside of the pressure switch. Possibly, it is a commonly available fitting from one of the major manufacturers such as Parker or Weatherhead (in the USA, anyhow).
I used a new pressure switch when I built up my compressor unit. I bought the kind of pressure switch which has a manual lever for working the switch contacts aside from the automatic operation. I think I spent about $30 US Dollars, but that was close to ten years ago. Another word to the wise: do NOT use hard plastic or nylon tubing for the unloader line. Compressed Air coming off the cylinder head is HOT. A lot of times nowadays, when I walk into supply houses and ask for various fittings for small bore copper tubing, the salesmen try to sell me the fittings and hard plastic or nylon tubing.
I would make a fairly hefty wager you are looking at compression type tubing fittings. It is unlikely that flared type tubing fittings were used in this application.