The club secured a deal worth £12m to bring the midfielder in from the Reds prior to the start of the Championship season.
Wilson has signed a contract until 2026 and was with the Reds since 2015, spending a total of six years at the club.
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He now has the chance to set down roots at Craven Cottage, though, and Crossley is delighted that he has made the move, having watched him regularly for Wales in the summer.
Wilson has won 29 caps for his country, scoring five goals.
Crossley told This Is Futbol: “He’s been playing a lot for Wales as well. I really, really like him and it’s his pedigree as well, he’s been at Liverpool for so many years, he’s done his little loan spells. What a signing.
“The kid’s a talent, he’s a talented player. I covered a lot of Wales games in the summer and he came off the bench quite a lot, but when he does, he excites you, he gets on the ball, he runs at people, he creates goals, he’s got a wonderful delivery from set-plays. Worth every penny.”