Fallows was NOT a technical director in the common sense, like Allison at Mercedes or Wache at RBR, overseeing all departments. Factually he was mainly head of the aero department, which his successor Eric Blandin (formerly deputy technical director - now renamed aerodynamics director) still is.
Fallows, Furbatto and McCullough all reported to CTO Andrew Green, until Green was sidelined. For a year there was no "strong man", overseeing all departments. Since 2024 these guys, or later their successors, report to Bob Bell, executive director technical. In about 3 weeks these guys report to CTO Cardile. Bob Bell is/was most likely the interim solution until their new structure is fully in place.
Fallows joins AMR after agreeing early exit from RBR (formula1.com 2022):
Fallows will form part of a triumvirate of technical leaders at billionaire Lawrence Stroll’s Aston Martin team, joining Performance Director Tom McCullough and Engineering Director Luca Furbatto – the latter having moved across from Alfa Romeo at the end of last season. The trio will report to Chief Technical Officer Andrew Green.
Bell joins AMR in 2024 (astonmartinf1.com):
He will have overall responsibility for the Technical, Engineering and Performance functions as the team continues its journey and development into true F1 contenders.