It is actually much more straightforward than it appears.
youve got 3 switches, a 3 core and earth between switch 1 and 2, and another 3 core and earth between switches 2 and 3.
Taking the terminal njumbers you have supplied.... which isnt waht we were expecting, TBH
Switches 1 and 3 are intermediates, but will be working as 2-ways.... Ignore terminal 4 on both these switches to make it easier to understand.... and terminal 1 as the two-way Common.
Switch 2 is the true intermediate.
take both browns in the 3 cores to the common in switch 1 and 3, and joint it through in switch 2.... do not connect it to the intermediate switch. Just stick them in a Wago together... nothing else.
Grey and black are the travellers.... or strappers, or pass wires... whatever you want to call them
on switch 1 and 3, the two ways... put them into 2 and 3....
on switrch 2, the intermediate .... put the grey and black from switch 1 into 2 and 3.... and the grey and black from switch 3 into 1 and 4
You now have to connect the live feed and the switchwire out.... which can be connected to either a grey or a black traveller at any of the switches... (but obviously not the same one at the same switch)