im sure you have this problem sorted now but two of the four red-brown wires will go in L2 of each switch then the other two go somewhere else downstairs. its possible the sparky who wired it was thinking it would be easier to take the power from the switch rather than to the light. its alot easier hanging a light when there is one cable insted of three. especially if its heavy. you could do with a tester really to find out where the wires go to.
its no good using a volt tester because the other two browns may not be live until connected to the proven live. most cheap multi meters have an ohm setting.
if u feel lucky try 1 wire at a time in L2 of the switch. when a light works try another in the other switch
when the two lights are working put the other two browns in common. doesnt matter which way.
if that works i'll eat my hair.
its the most likely answer from the description
it sounds to me as the wiring is fairly new because block connecting neurtals in switches isnt old school.