Managed to solve this myself with a live wire detector:
I believe the red/black on the far right of the diagram are wired incorrectly, as the black is live and the red neutral.
I believe this set up is described here, which states that it's not recommended for new installations for many reasons, but can be found in old homes.
Was it a single pole switch to begin with? Before running your own wire, there should have just been two wires tied to the switch (and a ground); the power coming in and the wire running out to the old light. The other wires should not be wired to the switch unless you want the switch to cut the items off as well (if they are downstream of it).
I drew up a quick, crude sketch of what you'll need to run this setup, regardless of what else is there. Get it down to the basics first and work up is the best way to troubleshoot.
Coming into your box should be the main hot line, I believe you said it was red. Pigtail off of it to the two commons on your new double switch and wire nut them all together.
Next, take your old wire that ran to your light and connect it to one of the other screws. This is where you'd decide which switch to control which item.
Take your new wire that you ran from your fan and connect it to the last screw. Then back at your fan/light, wire one of the lights to the fan and one to the light. Twist together and cap off the neutrals and you should have a working system.
I'm not certain of what those other wires are there for based on your description.. were they connected to the old switch? But, this will get the fan and light working.
Also, like mentioned in Iggy's answer, this won't allow you to control the fan speed from the switch, it'll just cut it on and off.
Best Answer
The OLD UK color codes are red for hot - and black for neutral.
The NEW EU color codes are brown for hot - and blue for neutral.
You have some of each era of wire in your house. I have no idea what happens after Brexit. For now we'll vote to Remain in EU color codes.
Get some brown and grey electrical tape
The reason to mark wire will become apparent at the very end, when the hookup instructions are simple.
The red wire with the piff of white paint on it is a hot-at-all-times wire. It needs to be marked with brown tape, e.g. wrap it around the wire a few times maybe 25mm back from the end of the wire.
The red wire with a U-shaped bend gets marked gray*. That is switched-hot to your older lamp.
The blue wire currently "marked" with crumpled brown tape... mark that gray. It is switched-hot to your newer lamp, though I am at a loss as to why they are using blue here - why not brown? What is the brown doing that's so important it can't be used as a hot? How does current to the lamp return? I sense trouble here. I would go over this work with a fine-tooth comb.
Now on the switch, this is a tricky switch since it's designed to be used as a 2-way (USA 3-way). Any 2(3)way can be used as a 1-way, simply by using common and 1 other terminal (not the third). Your switch has very terse labeling saying "in that case, don't use L2". So cover both L2 terminals with tape (just as a reminder, it doesn't serve any electrical purpose).
Now it wires just like your old switch. I would sanity-check whether use of a jumper is OK (to be more precise, landing two wires on one lug), that would either need to be generally allowed by UK/EU Code or specifically allowed by that switch's instructions (meaning it was tested/listed for that use). If it's not allowed, get a 4-terminal jumper block, create another brown jumper (buy more wire, don't cut this short thing in half), and put all four browns (including the red-taped-brown) into the 4-terminal block (including the 2 jumpers not yet connected). Regardless...
If you are very fastidious, you can put the browns to the COM and the grays to L1s, but on a simple switch it really doesn't matter.
* Using the new EU color codes for hot wires: brown, black and gray, avoiding black because it conflicts with your old-wiring neutral wires.