The word in use for the situation you describe is 'asynchronous':
1. Not synchronous; not existing or occurring at the same time, not coinciding in time.
In your examples, the use would be verbatim:
- This Catan tournament will include an asynchronous game.
- He never met his opponent in the asynchronous game of Uno.
Although the term 'asynchronous game' (or 'asynchronous gaming') is a comparatively recent coinage applied primarily to contemporary gaming conditions, it serves your purposes, and the purposes of others with the same descriptive needs, admirably. As Christian Dolan put it in a 2012 Vice article titled "Asynchronous Gaming Is Sexier Than it Sounds":
Asynchronous gaming has been around ever since Oxford dons first took a break from running Asian spy rings and having affairs with Vladimir Nabokov, and decided to play postal chess with each other instead. It’s as old, probably, as the Penny Black. Modern asynchronous gaming really got its mainstream start with Facebook, however, and its first star was Scrabulous.
More specific terms such as 'play-by-mail game' (often abbreviated PBM) or 'play-by-post game', along with 'correspondence game' have long been in use to describe chess and, since the 1960s, a board game called Diplomacy. The latter was, more recently, played via email.
Before being marginalized by computer-assisted gaming, a variety of play-by-mail games were popular:
In the 1980s, play-by-mail games reached their peak of popularity with the advent of Gaming Universal, Paper Mayhem and Flagship magazine, the first professional magazines devoted to play-by-mail games. (An earlier fanzine, Nuts & Bolts of PBM, was the first publication to exclusively cover the hobby.) Bob McLain, the publisher and editor of Gaming Universal, further popularized the hobby by writing articles that appeared in many of the leading mainstream gaming magazines of the time.
(Wikipedia, "Play-by-mail game")