The problem I see from the photographs you have supplied is that the three core and two core and earth cables (in the three gang switch) are not connected to the terminals in the way I would expect for a two way circuit. Hence as you suspect the wiring through the junction box has been mixed up.
I'm afraid you may have to go back to the beginning and trace the cables that come into the three gang switch back to the their point of origin.
Split the process into two stages dealing first with the chandellier and wall lights and focus on the three gang switch wires.
At the three gang switch identify the three core cable. This will be red/blue/yellow and earth. Under normal circumstances these cables would act as 'strappers' and provide the two way link between the chandellier/wall light switch in the three gang and the single gang. Pick one of the switches on the three gang as your chandellier switch. You would put the red in the Common, the blue (with red sleeve) in L1 and the yellow (with red sleeve) in L2 on both switches.
Identify the two core cable. This should be red/black but from the photographs it could be red/red. You need to identify which on these two cores is live and which one is switch live (the one that goes to the live at the lights). Once you have identified which is which the live goes into L1 on the three gang chandellier switch and the switch live goes to L2 on the same switch.
This should enable the two way switches to operate as required.
To get the other two switches on the three gang to operate as normal switches you would have to run two twin and earth cables from their respective live power sources to Common on the second and third switch and their respective switch lives back to their lights.
There is another way to do this by linking the commons in the three gang switches to give a permanent live to switch 2 & 3 then run switch lives back to the appropirate ceiling rose light.
You can see how difficult this is to explain from afar - especially if there is a junction box in the way and the wiring is less than normal.
You could do with a multi-meter to establish which cable is live and switch but you would need to understand how to operate one.
If all this appears too daunting then I would suggest you get an electrician in to try and sort out the mess.
A picture of the cables at JB area marked with what you have already identified would be good.
Sorry River I understand the direction you are ultimately heading (the same as me with the common live) the OP is trying to restore the system back to now it was originally working
(common live). There is no need to run any new cables, I think what you are initially describing may confuse the OP more than he already is.
edit to clarify/improve wording