Job done. Easier than I thought. For anyone wanting to do this:
The two ground cables were both copper coloured, the left was blue and the right red.
I used a multimeter to work out which connector on the new TRS connector represented the Tip (Left), Ring (Right) and Sleeve (Ground), I suspect most connectors are the same. The large connector in the center was ground, the middle one left and the bottom right. I cut the cables, removed the enamel insulation with a cigarette lighter and soldered the wires to a TRS connector I bought for a couple of quid.
My headphones live on to another day and I don't have to cart about the RE-ZERO TRRS-TRS adapter anymore.
I recommend this fix to anyone; it's a bit fiddly because the cables are so small but it's not difficult and there's few things that can go wrong as you can always cut the wire and start again. You need: a new 3.5mm TRS phono connector, cigarette lighter or some fine sandpaper (to strip the insulation from the wire), soldering iron and a pair of scissors or a sharp knife (to cut and strip the wire). Takes about 10 minutes if you haven't done it before.