I would cut it all down to at least 2-3ft and start again with wires and ties. As you have 3 plants in there try to fan out the outer two, keeping all 3 plants fairly spaced. You will need to prune back every year either stopping the plants from interlacing or allowing them to do so, as you wish. After the winter jasmine has finished ... about now, or before March ... would be a good time to cut them all down.

Otherwise, drill holes in the wall, and put in galvanised eye bolts with fiber rawl plugs. Once the cement in the plugs has set, tie the trellis to the eye bolts.

I can't even see a trellis in there. I'd cut all the growth off what's there already and fix a rigid trellis fencing panel (squared sections, not criss cross) of an appropriate size to the wall. These are much stronger and will be capable of supporting the weight of the plants. Remember to fix it so that its slightly away from the wall by putting it the right way round - with the vertical struts on the edge of the panel flush to the wall, meaning the rest of the wood is about an inch or so away from the brickwork.

You can see some of the green plastic mesh at the bottom mixed in with the stems. I'll get onto the cutting back tomorrow if it's not raining. Thanks everyone!

Ah, well if its green plastic mesh, you may not need to lose all the topgrowth. By cutting the mesh in various places, you can extract it without losing to much of the plants.

Hmm, what I'm seeing is one of those cheap PVC or polystyrene trellises they marketed in the late '90's and early Aughts. A little harder to cut than mesh, but a pair of loppers should do the job.

Too late! I've cut back the rose and extracted it from the plastic mesh, and removed all I could of the honeysuckle as I found it had spread and rooted from a bit further along. We need to get a masonry drill, some sort of securing system and a longer electrical extension to finish the job, but at least it's not still annoying me whenever I look out of the window. Thanks for all your advice.