Wyse is a first-generation Irish American, and was born in Queens, New York. He discovered the music of Kiss, The Doors, and Led Zeppelin in grade school, and was inspired by Iron Maiden's Steve Harris to play bass. Performing with local bands in New York, Wyse's ability and innovative approach to the bass was quickly acknowledged, and at 17, he was profiled in Guitar Player Magazine. At 20, he was featured in Guitar For the Practicing Musician.
In 1990, prompted by the success of a studio session with musicians associated with Steve Vai, Wyse moved to Los Angeles, where he devoted most of his time to playing with bands. After performing with artists including the band Lusk featuring Guns N' Roses's Chris Pitman and Tool’s Paul D'Amour, Wyse was introduced to acclaimed producer Bob Rock, and became Rock's 'go-to' bassist, playing with Rock-produced artists including Tal Bachman, as well as continuing to play with other artists, including Scott Weiland.
In 2001, Wyse played on The Cult's well-received album, Beyond Good and Evil, which Rock produced. In 2006, he became The Cult's full-time bass player, touring worldwide and recording two additional albums.
In 2003, Bob Rock, who was then producing Metallica, urged Wyse to audition for the band. He was one of a handful of bassists selected to play with Metallica during the audition process, which was documented in the film Some Kind of Monster.
In 2004, Wyse was recruited for Ozzy Osbourne's band by drummer Mike Bordin after they met while playing as part of Jerry Cantrell's solo band. He played on Ozzy's Under Cover album, which was included on Osbourne's box set, Prince of Darkness. Wyse also played on Jerry Cantrell's covers band Cardboard Vampyres. Additionally, Wyse performed and recorded with Bob Rock's band, the Payola$.
In 2007, Wyse founded the current incarnation of Owl, an alternative rock band. In Owl, Wyse acts as singer, songwriter, and producer, in addition to playing bass guitar and upright bass, sometimes creating Hendrix-like effects on the upright bass by using a bow. Wyse formed Owl with his childhood friend, drummer Dan Dinsmore, and LA-based guitarist Jason Achilles Mezilis.
Wyse worked the on the third Owl record, Things You Can't See, in 2014. In addition, he played bass guitar on Ace Frehley's album, Space Invader  He toured with Frehley following the release of Space Invader. Things You Can't See was released in 2015, as was Frehley's Origins Vol 1. Wyse played bass on the record, which featured Paul Stanley on vocals for the track Fire and Water.
Wyse left the Cult in 2015, and has continued to tour and record with Frehley. In January 2018 he played "New York Groove" with Frehley prior to the Bridgestone NHL Winter Classic between the New York Rangers and Buffalo Sabres at Citi Field. The song is a staple at New York sporting events; it is played when the New York Giants score and when the New York Mets win.
Wyse joined the Hollywood Vampires in 2018. A supergroup that includes Joe Perry, Alice Cooper, and Johnny Depp, Wyse began a world tour with the band in May.