Allsop started his career as a youth player at West Bromwich Albion, before transferring to Millwall in 2011. After short spells at Icelandic club Höttur and Leyton Orient, he joined AFC Bournemouth in 2013. He spent some time on loan at Coventry City, Wycombe Wanderers, Portsmouth, Blackpool and Lincoln City, and made 23 league appearances for Bournemouth, including two in the Premier League, before returning to Wycombe Wanderers in 2018 for three seasons.