Depends how much you want to spend (but I expect you know this).
There are post production houses in Soho and elsewhere that have 8mm gates for telecine ("TK") machines. That would (by far) give you the best results, but you'd pay a commercial rate. If you can present the film on one large reel, it might be cost-effective, if the material is valuable. You might need a wet-gate machine (to reduce scratches), but that in turn may barf at tape joints. Well-made cement ones would be OK (probably!).
Most semi-pro stuff on 8mm stock was super-8 format. If the TK machine can't do standard-8 you'll get a rough black border, but that may not matter. It's fairly easy to replace with a clean matte in modern video editing software.
I think there are still two commercial TK channels in Bristol, because of the Nat Hist connection. BBC post Production had one, Films@59 I think had another, and there was at least one more. The BBC one definitely had an 8mm gate and I'd expect the commercial ones do. Demand has been falling off though, as most of the BBC Nat Hist library is now digitised.
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DIY: If you Google "8mm telecine" there are quite a few hits. One guy is using a Canon 5D II camera to do "HD" transfers, and another has converted an Eumig projector. It's fairly easy to focus a camera on the film, the difficulty is to get even illumination, without a "hot spot" in the middle. If you use a lamp, you will need to set up the collimator carefully (put a sheet of greaseproof paper in the gate where the film goes, and adjust until you can see the lamp filament focused on the paper, usually - watch out as it burns easily, and sunglasses are helpful!).
The most annoying issue though will be the frame rate: European TV is 25 frames/sec (fps), US is (almost) 30. Commercial film is 24 fps, and historically was run 4% faster in Europe for television, so that there was frame to frame equivalence. Unfortunately, amateur 8mm was usually 16fps, rarely any faster, and it won't transfer very nicely to 25fps without some posh electronics. Computer frame rates go down to 16fps though, so you might transfer to one of those formats instead.
Super16 on a good day is equivalent to HDTV, but any film format smaller than that really isn't. There is an argument that you don't make the grain any worse going to HD, but it's probably not worth it. That said, commercial TK channels will almost certainly be HD these days.
Finally, if you go to a non-broadcast facility, ask for their show reel of similar material, and give them a test roll, if you can, but not of something especially valuable, in case they're rubbish and scratch it.
Probably too much info, but hey.
E...
... a senior colleague of whose (long, long ago) won a film sound BAFTA last Sunday for Les Mis.
