On this day in 1962, John Hershel Glenn, Jr, became America's first man in orbit*. https://en.wikipedia.org/wiki/Mercury-Atlas_6 That doesn't look much like a Speedy though. (https://www.nasa.gov/content/climb-aboard) (* As correctly pointed out below, he was not our first man in space)
First American in orbit, Alan Shepard was the first American in space. Great picture, I can't tell what the watch is, maybe someone who is better at this will tell us.
It actually looks more like a gauge than a watch, though what it's connected to at that point is anyone's guess.
It's a Heuer ... too tired to remember the model ... but Google found this link: http://wornandwound.com/watches-space-john-glenn-heuer-2915a/
That does look kind of deep for a watch but on his right wrist there appears to be a watch. That must be the Heuer.
Here’s a better picture. You can see the Heuer on his right wrist. And here’s a gratuitous picture of my personal Heuer 2915A
As I read the title I was thinking the topic was about the Blue Origin "New Glenn" reusable launch vehicle... scheduled for 2020 Anyway, there's only one good source for time-period spaceflight wrist watch photos . Mercury astronaut John Glenn wearing the TAG Heuer stopwatch .