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If your tires were 26×3 beaded edge (Clincher/Wulst), you cannot put more modern straight side or well-base tires on without installing new and different rims, which would be a pity. If you have sufficient air pressure in beaded edge tires they behave very well. I have seen Bugattis win on the Silverstone circuit with beaded edge tires with a quite coarse chevron type thread.
A 26×3 clincher tire will fit on a 20″ beaded edge rim. No other tire (except perhaps 27 x 3.85) will fit. The Coker tire will be OK for touring use, and probably also on the track. The Dunlop thread is not bad either, I have used the same thread on 28×3 tires on Harleys from the early 20s with no problems. The only problem with beaded edge rims is corrosion. If the rim edge is corroded, it may be very sharp – like a knife – and cause damage to the tire and punctures. Too low pressure may also cause the tube to move about and cut the valve stem.
However, I wish you luck with your project!