Weller has been chief financial officer of G4S' since 2016. The security firm is in the process of being taken over by U.S. rival Allied Universal after a majority of shareholders in March accepted the 3.8 billion pound ($5.36 billion) deal.
Gordon Boyd, who has been interim CFO of London-listed Capita since November, will leave on Thursday as planned, the company said.
Weller will join Capita on Wednesday and will be an executive director on its board, it added.
"Tim's expertise will be invaluable in leading our finance team and supporting our ongoing transformation," Capita Chief Executive Officer Jon Lewis said.
Capita, a provider of consulting and digital services to public and private sectors, launched its recovery plan in March after the COVID-19 pandemic disrupted its previous push to return to sustained profit and growth.
Weller has nearly two decades of experience in top finance roles, including stints at Innogy, Thames Water, United Utilities and Petrofac, Capita said.
($1 = 0.7084 pounds)
(Reporting by Pushkala Aripaka in Bengaluru; Editing by Anil D'Silva)