It may be necessary and almost certainly will be safer to hire a qualified electrician.
The first step in diagnosing this problem is to identify the conductors.
You have a multimeter, but you should check that your leads, probes, and the meter itself are rated for mains voltage. Switch off the entire electrical supply at the consumer unit or main isolator switch. Use a voltage tester or--at a minimum--a neon test screwdriver, to verify that each conductor in the back box and at light fittings is dead. Do not conduct the following test on a live installation. Turn off all circuits.
One way of identifying conductors is to connect the live and earth and/or neutral and earth wires of a single cable together using a terminal block or similar at one end, then at the other end, test continuity between the L+E or N+E with your multimeter. You can discount the possibility of an existing short by testing continuity between the conductors before connecting them together at one end.
In your case, you need to know which of your red and black wires go to/from light fixtures, and which are live or neutral feeds. Given the age of the installation, it is unlikely you will have a neutral at the switch.
You may not be able to identify the additional white-sheathed twin and earth cable in the back box, so strip it back and terminate each conductor. Do not connect them together, do not tape them, and do not leave it unterminated. If you can identify it, and find that it is dead, you may be able to cut it back or remove it completely, thus avoiding future confusion.
Once you have identified the conductors, you will be able to connect a switch according to the manufacturer's instructions.