Wade quit Gallagher Premiership side Wasps in October to join the NFL's International Pathway Program, which offers players from outside America a route into the sport.
The 27-year-old was expected to be included in the NFL Draft later this month but has instead signed for the Bills, who finished with a 6-10 record last season.
"With the recent news I'm completely lost for words," said the former Wasps winger. "Every week that has gone by has exceeded my expectations.
"I didn't really know what to expect from the process and have tried to stay open to all possibilities. I've tried to take every moment as it comes and hit it hard."
Wade, who has been identified as a running back by the Bills after scoring 82 tries for Wasps in the Premiership, added: "Like I said, at this stage I'm speechless.
"I've put a post on Instagram but couldn't come up with a caption for it.
"I've had a lot of messages from close friends and family but I'm still trying to get my head around it.
"For me this is another step closer to my dream and what I set out to achieve from the start.
"It's making me emotional thinking about the risk I took and the magnitude of the move. This has made that even more apparent. Every time I think about, I start to well up."
The Bills will get a roster exemption from the NFL for Wade through the International Player Pathway Program, allowing them to carry 91 players instead of the typical 90-player roster.
Wade, third-highest try scorer in Premiership history in England, participated in an NFL Pro Day last week and recorded a time of 4.53 seconds in the 40-yard dash.
He follows in the footsteps of former Worcester Warriors forward Christian Scotland-Williamson, who was allocated to the Pittsburgh Steelers in 2018 before being signed to the team's practice squad.
Scotland-Williamson went on to sign a reserves contract with the Steelers at the beginning of this year.
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