The fuse protects both wiring and solenoid, but mostly the solenoid. Possibly less important now, as new solenoids are a lot less expensive than they used to be, but still a good idea, IMO.
I use an MDL 8 fuse, which is an 8 amp "slow blow", mostly because I had a case of them on hand (discarded by a previous employer). But I believe even an ordinary 5 amp fuse (ie AGC 5) would work fine. Or if you want to use a more modern fuse holder, ATO or ATC, etc would work fine too.
The solenoid draws a big burst of current at first (hence the relay and relatively beefy wiring to it), but normally the burst of current only lasts a fraction of a second; not long enough for even a 5 amp fuse to get hot and burn out. The problem is that if something blocks the motion of the solenoid plunger and it doesn't cause the internal contacts to open, the high current will eventually (in a few minutes) cause the solenoid to overheat and burn up. So using a smaller fuse means the fuse blows before the solenoid hurts itself, giving you a chance to correct the problem without needed a new solenoid. (In my case, it was a bit of gravel that had worked it's way inside and jammed the plunger.)
The ratings on the diode are not critical at all; what I used is a 1N4004, which is rated 400 volts and 1 amp. (At the time, Radio Shack sold a bag of 25 of them for a couple of dollars, but they don't do that any more.) Any of the 1N400x series should work fine, and they are readily available at any electronics supplier (Digi-Key, Mouser, Newark, etc) or on eBay. Just for example
https://www.ebay.com/itm/25-x-1N400...095096?hash=item3cb99cbe78:g:ml0AAOxyFiRRzFEx
The one Geo showed above will work just fine too, if RS still has them (and if there is still a RS available to you. AFAIK all the ones near me are gone now.)
If you happen to be near Torrance, California (or passing through), I recommend Torrance Electronics on Carson Ave. All the other stores in the area have closed, but TE was still there last time I drove by.
Note that the diode is polarity sensitive, installation will depend on whether your TR3 is still positive ground or has been converted to negative ground. The diode will usually have a white stripe on one end to indicate polarity, the stripe needs to go towards the positive side. So for positive ground, it would go towards ground; for negative ground it goes towards the relay output pin.