Firstly, I wanna say a huge thank you to everyone who's contributed to this thread. I've not quite sorted it, but I think I'm almost there, learning and doing things I didn't think I'd be able to do in the process.
I took
@mcshroom 's advice and purchased the 7 speed cassette, spacer and got a KMC Z7 chain.
After cleaning and degreasing the whole bike, I got the cassette on, fitted the chain and installed the chain yesterday (when chain breaking tool arrived). Image before the chain was broke and joined:
I woke up early this morning to see if I could get things indexed and working correctly in order to use on my commute, however I've created another issue for myself. I mistakenly thought the rear derailleur's barrel adjuster was what needed to be turned to change cable tension, however there's an adjuster that does that up the frame closer to the bars. Through trying to turn the derailleur's adjuster, as it's so old and worn, it snapped, leaving part of itself inside the thread on the derailleur. Everything still looks the same as the image above, but the barrel adjuster can now be lifted away from the derailleur.
I don't know how much of an impact on shifting this will have, but it seems the adjuster near the bars is still doing it's job (changing the tension of the cable). Knowing it's broken though, it's something I'd like to attempt to fix, however I can't find that same barrel adjuster part anywhere online, does anyone know of any alternatives?
Will refitting a new barrel adjuster be a big job? I'm just wondering how I'm going to remove the cable before trying to remove the broken piece if part of the fitting is blocking the cable's path.
I'll take a closer look tonight and post in some new images.
Thanks again.