Gleeson, the 30-year-old Northamptonshire quick, has only been playing the professional game for three seasons having previously plied his trade for Cumberland in the Minor Counties set-up. In 2016 he had a match-by-match deal with Northamptonshire, largely due to the county's difficult financial position, but impressed enough to earn a three-year contract although little more than a year ago was still working as a community coach.
"I was sat having dinner out with a few of the lads when James Whitaker rang. Rips [David Ripley, Northants coach] had said that Andy Flower had been in touch on Thursday night to enquire about my availability, but to actually get the call was just a phenomenal feeling," Gleeson said.
"Just over a year ago I was working as a community coach up at Lancs so to get the opportunity to play for my country is a dream come true, though something I never thought could actually happen. I'm over the moon."
He was already due to head out to Barbados next month for the North v South one-day tournament, but now has the opportunity to get immersed with the England system at a time when his name has been touted as an option to bring some extra pace.
Gleeson has a strong record across all three formats: in first-class cricket he averages 24.91, one-day cricket 30.27 and in T20 22.51 with an economy rate just over seven an over.
Last season, his coach David Ripley endorsed Gleeson's T20 credentials for England although admitted that was not the format the county had first earmarked him for.
"The white-ball stuff was actually a surprise because he had caught the eye in red-ball cricket," Ripley said. "He had a little bit of pace and could hold the ball up a little bit from a slightly angular action and we liked the look of him.
"But then he got into the white ball team and the speed he bowled with the white ball in short spells, to which he has added a yorker which he has improved this year, means he now looks like a banker in all formats."
Also speaking last summer, Gleeson reflected on his belated journey into the professional game,
"What had started out as a six-month position with Lancashire Cricket Board had turned into a full-time job, doing a lot of Chance to Shine work with schools," he said.
"I worked in a factory for a few months after university, so it was great to be working in coaching, which is what I'd set out to do with the course I chose at university.
"But when I was playing in a match for Cumberland against Bedfordshire, their captain, James Middlebrook, told me he thought I had something and said he could put me in touch with Northants. They played me in a Second XI game and then threw me in against the Australians in a tour game. I got a couple of wickets and it went on from there."
His Lions chance has come after Garton, the Sussex left-arm quick who was drafted into the Ashes squad as cover early in the tour, suffered a side injury in preparation for the Lions warm-up match against a Jamaica XI. He played in the game, but managed just five overs - taking 2 for 13 - and has now been forced home to recover ahead of the domestic season.
Gleeson won't arrive in Jamaica in time to be considered for the first four-day match against West Indies A which begins on Sunday but will be in the frame for the remainder of the tour.