I am far from a Triumph expert but over the years I have owned numerous bikes. Anything that has been sitting for 20 years deserves some very special treatment... carburetors in particular most probably are gummed up inside and at least contributing to your problems. I would remove them, break them down and at least soak them in lacquer thinner for a day or two. Others may know of some more potent liquids & treatments. This may or may not solve your starting problem. But it is certainly something worth doing to help bring this machine back to a baseline situation.
