I have two zone valves and two pumps, and it needs relays to make it work correctly, under floor heating controllers often have relays built in
but with set up you have likely bespoke, where some we hope cleaver heating engineer who installed it has designed a system.
So first question, has it ever worked?
So in my case the zone valve has a micro switch, so as it opens it can switch some thing else on, for example boiler and pump, or boiler, or pump, but if boiler and pump, then that means both boiler and pump are connected together, as only one micro switch.
So in my case the micro switch turns on a relay, this relay has three sets of contracts, however only need two sets, so it switches on both pump and boiler, but when not activated boiler and pump not connected, so another relay can turn on same boiler but a different pump.
I can't see any device which would allow the two boilers to work without some sort of relay.
I am lazy, and if I had to do that job likely I would use a PLC to control it, as I know how to program PLC's so for me easy, but how a plumber or heating engineer would do it not sure.
I would start with a truth table, I would guess.
1) If no fire lit, water must not circulate, as hard to light a fire if water cooling the fire.
2) Once fire lit, water must circulate.
3) Likely a three port valve like used in W Plan, so if rooms warm enough water goes to sink radiator to simply get rid of heat, and other wise circulates through house radiators, but could also be a by-pass valve.
4) Some method to ensure flow even if pump fails like a power cut, sons system used 12 volt battery backed pump, but could also be thermo syphon to heat domestic hot water (DHW).
5) Relay on pump to stop oil boiler firing if back boiler heating water.
My brother-in-laws old house had similar system, he had twin storage tanks something like this
and a very complex control system to ensure always spare capacity for hot water from solid fuel water heater.
Going back in history mother house would boil the water, many times I would be sent to run off hot water, as it was boiling. Noise told us what was going on, how this happened killing Rhianna Hardie I don't know?
may be insulation stopped the noise normally associated with water boiling.
The main point is it should fail safe, be it sink radiators to get rid of heat, 12 volt battery backed pumps, thermo syphon, or other method, we see adverts for very cleaver stoves
but very little on how one ensures the pumps don't fail or methods to put fire out should pumps fail.
Gas and oil is easy, remove power to oil or gas valve and boiler stops. Solid fuel is not so easy, I remember my mother raking out fire onto a shovel because the water had been turned off in the street. Which is why quarry tiled floor, hot coals would not damage it.
Same with sons narrow boat, tiled floor under stove so fire can be raked out without worry of setting floor on fire. Clearly raked out onto shovel, but can spill coals.
Personally I would want some one trained how to control solid fuel fires, too easy to miss some safety feature.