Bedwell joined Swindon Town as a child, making his way through the youth ranks with the Wiltshire club. Bedwell was given a first team squad number at the age of seventeen - when still a part of the youth set-up - acting as back-up to deputy Phil Smith, when first choice keeper Mattia Lanzano and third choice Mark Scott were both injured at the start of the 2011/12 season. The arrival of Wes Foderingham prevented Bedwell from keeping his place in the starting sixteen - but at the end of the season, though Smith, Lanzano and Scott were all released, Bedwell was offered his first professional contract, and he signed a one-year deal.
On 2 September 2012 Bedwell made his first team debut for Swindon in the 4-1 loss at Preston North End. With Swindon losing 2-0 early in to the game, Town manager Paolo Di Canio decided to substitute first choice goalkeeper Wes Foderingham for the rookie Bedwell.
Bedwell was loaned out to Banbury United September 2013 in order to gain first team experience.