Baggaley moved to the Birmingham City academy in 2013, a year later she was promoted to the senior team as a backup to fellow One Glove pro Becky Spencer, named as a substitute for all 14 matches. She made her senior debut on 1 May 2014 in a 2–1 win over Yeovil Town.

Following the departure of Spencer to Chelsea, she began the 2016 season as the undisputed starter. Baggaley then joined Bristol City on loan in 2017 and later joined them permenantly. At the end of the 18/19 season, she won the FA WFA Players' Player of the year after 6 clean sheets which helped Bristol to 6th place, their best finish since 2013.

In July 2021, Baggaley signed with Manchester Utd and made her club debut on 14 October 2021 in a 2–2 away draw with Championship side Durham in the 2021–22 League Cup group stage opener. Manchester United took the bonus point with a 5–3 penalty shootout victory.

Recently, Baggaley joined WSL side Brighton & Hove Albion Women in September 2023.