Before you remove the rear chain, get another chain which can be attached to it and used to draw it off the gearbox drive sprocket, thereby leaving a chain in place: this makes replacement much easier.
Worn-out chains will feel very floppy when you have them off the motorbike. The standard formula, as mentioned here, dictates renewal when stretch equals 1/4" per foot or 6.5mm/30mm. Just lay the chain on the workbench beside a ruler, push the links together, and then pull them apart. A worn chain will also bow significantly sideways.
The same applies to the primary chain. Comet primary chains can live hard lives if the machine is used mainly for short trips around town because the ESA - the spring-loaded ramp cam shock absorber on the crankshaft - is more robust than, say, that found on BSAs and Triumphs and it can be as if the machine doesn't have one. Check the chain rollers for signs of cracking.
Sidenote on that score: have a good look at the tabs or 'ears' of your clutch friction plates. They take a lot of abuse because of this and I have known them to be ripped off new plates on a single long road trip, creating all sorts of hassle when trying to control the machine through towns! Remedy is to fit "big ear" plates with a basket to match or, as I shall be doing, a modern clutch mated to the chainwheel. I don't want to spook you or anything, as a new Comet rider. Just giving you a heads-up based on using a Comet pretty much as an everyday nail.
Chain adjustment: good points made here about chain tension. Get the Comet up on her rear stand and turn the rear wheel around, feeling the chain tension in the middle of the upper or lower run. There will be places where the chain is tighter than others. Always adjust a chain at its tightest spot. Ideally, you should have a friend sit on the bike but if you have no friends, then make allowances by increasing the 3/4" figure for the rear chain to and inch or just over, say 1 1/4". Better slightly too loose than too tight because a tight chain is no good for bearings and can even damage bearing housings in extreme cases. Remember that your hubs are made of alloy and that there is a reason why so many Vincent-HRD rear hubs are only useful as paperweights!
Likewise, find the tightest spot in the primary chain. The gearbox pivots on the upper mounting. The adjustment bolts run through the housing, with locknuts, and bear on the lower mounting. You'll need a good cranked wrench to slacken off the nuts (effected on the timing side) and, moreover, to tighten them up again. Just slacken off the nuts. No need for everything to be flopping about. Loosen off the adjusters' locknuts and the adjusters. Check chain tension. The requisite slack is 1/2" at the tightest spot. It will probably (hopefully!) be too loose.
Use the front adjuster to tighten it appropriately. Tighten the rear adjuster snug against the pivot. Nip up the locknuts. A point to note at this stage is that a gearbox will always want to move backwards when the motorbike is rolling along, even if most people might assume the opposite to be the case. Given the inevitable wear and slack in decades-old parts, it is always good policy to ensure that a pre-unit gearbox is "pushed forwards" when finally adjusted, thereby taking up any slack in, for instance, the mounting lugs, in which the axial mounting studs might be a tad loose sixty years on. In this case, your main 'player' is that front adjuster: if your final adjustment is done on the front, the gearbox will be prevented from creeping backwards as you unleash the horses.
Now kick the engine over a few times. Recheck the chain tension and keep working until you get it right. Recheck the chain tension after doing up the gearbox mounting nuts dead tight as this can affect tension in some cases, depending on wear of components. Once you have adjusted the primary chain on any old pre-unit machine a few times, it becomes an easy enough task although it can be irritating. Then adjust up your rear chain and you're ready to rock and roll. Regarding lube in the primary chaincase, I tried all sorts of things but I am using simple two-stroke oil at the moment, with good results. Paris is not the best place for finding monogrades, fork oils are massively overpriced and the 20/50 stuff I use in the engine is a bit draggy in a Burman clutch, although my TR5 Trophy is happy on it.
Hope this helps. A well-sorted Comet can be a very fast machine. In fact, I would put it up there with Norton Inters in terms of long-legged fast touring. They were sneered at for a long time as "half a Vincent" but the postwar Comet is far closer to Irving's original vision than the veetwins. Not that I don't love 1000s. I have one. But get out there and enjoy your Comet. They're pretty damn good for British singles of the 1940s!
Prosper Keating
Paris