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On relection I have two used but useable tan wall Michelins (1 Speed, 1 zigzag) which I would be happy to swap for serviceable black tyres, preferably Schwalbe. If your swaps are better than mine (e.g. new Schwalbes) I'd be happy to make a cash adjusment in your favour.
Getting tyres made would be great if it could be done, but I fear it would involve a large and very risky financial investment; basically a sledge hammer and nut situation. If I really thought it might be feasable I would try to involve the Veteran - Cycle Club and my first port of call would be Veloflex, who make really excellent tyres on a relatively small scale.
Another possibilty for your machine would be to use sprints and tubs, which certainly existed when your bike was new even though Dawes would never have fitted them (although owners might well have done). There are still plenty of rims lying around that nobody wants, and you should be able to find 32/40 spoking to match your hubs. I don't know what heavy tubs cost nowadays, but compared with getting tyres made they would be cheap, and you could certainly afford to throw away any that punctured. As a bonus sprints give an excellent ride.
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I was talking to Rikki at Pankhurst Cycles a while back, and he has several 26x1 1/4 tyres. You can find him here: https://www.pankhurstcycles.co.uk/
Thanks to all for the replies re. my Dawes. I have been wondering. Would it not be possible, somehow, to put it around on all forums, facebook groups etc etc. and gather together sufficient interest to put in an order for a short run of decent tyres in 26 x 1 1/4" tan wall with, say, Schwalbe, Panaracer, Vittoria or even Chan Sen (?) Surely SJS would take a few? Other online shops? Etc...?