The way this is normally done, eg when a Hive system is installed, is to
removed the timer. On modern systems ( and that can mean several decades
old) the timer will probably be mounted on a Mark 14 back plate. This has a
footprint the same size as a single mains socket etc. There is a row of
terminals - as shown in the right hand group of your diagram- and the timer
/ programmer clips onto it.
Generally, to convert to a remote control system - such as Hive- you unclip
the programmer, clip on a box which has relays ( or equivalent) plus the
remote control interface. Sometimes minor adjustments are needed to the
existing wiring but the theory is, the Mark 14 plate provides a common
interface.
You should be able to adapt the idea.
Simply fit a relays wired to emulate the switching arrangement in the timer
part of your diagram, removing the existing programmer and connecting your
circuit to the corresponding points.
The coils of your relays will be powered by your control circuit - I assume
an Arduino or R Pi.