Councillor Fage was elected unopposed at the town council's annual statutory meeting, having served as the town's deputy mayor for the last two years.
Councillor Russell has served a three-year term as the town's mayor guiding the local authority through the pandemic.
Proposing councillor Fage, who was elected to the council in March 2017, councillor Hazel Ramsay said: "He's done a wonderful job as deputy mayor and deserves to step up."
A former Stratton Grammar School pupil, councillor Fage told the council: "It's a tremendous honour to accept the position of Biggleswade mayor and represent our town at such a crucial time.
"Over the coming decades, this town will face enormous change," he explained. "The choices ahead of us, as councillors, and the decisions we make will shape Biggleswade for a very long time to come.
"We take our duties seriously and make decisions carefully, caring deeply about our town.
"This council has a strong team of councillors and staff from a wide range of backgrounds with diverse talents. I look forward to working with the team in this new role in the best interests of Biggleswade.
"Someone mentioned to me becoming mayor, especially while in my 20s, is an accomplishment. While there might be a little truth in that, for me it means rather little compared to the ambitions we have as a council for our town.
"There's the delivery we've started and the genuine accomplishments we can attest to, which are also to the benefit of our residents.
"In councillor Russell we've been blessed with an outstanding mayor," he added. "She's seen our town and our council through truly exceptional times. Her drive to deliver during the last three years has been clear for all to see.
"Her steadfast delivery of the neighbourhood plan, which will shortly go to referendum, and extremely detailed response to Central Bedfordshire Council's Local Plan and the development of the town council's staff team are all immense achievements.
"This will put Biggleswade in a far better position than perhaps sometimes we realise. I love this town and look forward to working hard for it. And I formally thank councillor Russell. The town owes her a huge debt of gratitude."
During Tuesday's (May 10) meeting, the mayor and his deputy read out separate oaths to fulfil their duties "to the best of their judgment and ability".
Councillor Fage put on the mayoral chain, which was unavailable at the corresponding meeting 12 months ago as it was undergoing repairs.