I've been looking for the answer to this question for a while now and I seem to have narrowed it down to three candidates. Those candidates are John (or is it William?) Blakey, Richard Trevithick and Oliver Evans. I'm not entirely sure about the details of Blakey's boiler beyond the fact he did use cylinders, though they were very small and I'm not sure if the case in which those cylinders were held was cylindrical or not. Richard Trevithick and Oliver Evans seem to have designed a cylindrical boiler at roughly the same time but different books give different dates, with one book claiming Evans invented the cylindrical boiler as early as 1792! Others claim Trevithick was first in about 1804, with Evans following shortly after. I've yet to see any really conclusive proof that any of these people was first so I was wondering if you guys could help me out here.
Trevithick and Evans were working about the same time on a cylindrical high-pressure boiler, and knowing nothing of each other. If you want to say it was invented when somebody just started to talk about it or tinker with the idea, Evans, Trevithick or even William Murdock would be in the running. But Evans' boiler actually being in use can be dated reliably to 1805, while the date for Trevithick's is more uncertain- definitely in use before 1812, but how long before?
Willaim Blakey's boiler is of another kind called a water-tube boiler, which would later become a common type, and he was indeed quite early with the idea, in 1776. But he tried to make the tubes with brazed copper tubes, and they failed. On the other hand, James Rumsey did actually create a strong water-tube boiler that worked for his 1787 steamboat, perhaps because had a gunsmith weld up and bend the iron tubes for him, and then screw them together with threaded iron unions. When he relocated to England in 1788 he soon put out a patent with lot of different boiler designs, but he died a few years later and so few if any actually were made.
Though the water-tube boiler could be stronger than the nesting-cylinders boiler of Evans and Trevithick, it should be noted that Rumsey and Blakey only wanted to build low pressure engines of the condensing type. The steam pressure in their engines was around 14 psi, whereas Trevithick and Evans were trying to run engines at closer to 100 psi. Rumsey saw the water-tube design only as a way to make a boiler small and light enough to be on a boat.
Dickinson: A Short History of the Steam Engine