I'm surprised the last switch in the kitchen has a BLACK going to a common, the common which feeds the other switch common.
I am wondering if this is stopping a live feed. Try leaving the one way switch in the on (or off if it happens to work upside down) position, and connect the middle switch with the single reds to the commons, a pair to L1+2, other pair to L1+2. Swap the commons over if doesn't work.
Appologies if it doesn't work. Work safely.
It's not easy without a multimeter to trace the wires.
What does that one way switch at the last switch do? I don't think you answered that earlier. Outside light?
Either Ban or Coljack (can't remember) has a wiring diagram of two way switching done with two core cables (as opposed to the usual 3 core cable) which may be of use to you.
With a multimeter you will be able to find the permanent live, live to the light, and the 'strapper' wires which bridge the two way switches.
Do you know how to use a multimeter?
Please remember to work safely and carefully.
I presume just one fuse or circuit breaker feeds all these downstairs lights?
Remember the upstairs switches will probably be on a different fuse or circuit breaker.