It looks to me like it could be one of the evergreen clematis, but I could be wrong.

it reminds me of 'kangaroo vine' which is a houseplant for me in the east yorkshire region.

Definitely clematis foliage in the second picture. Kangaroo vine or grape ivy would have alternate leaves, instead of the opposite leaves shown.

If nobody gives you a definite identification I would leave it for a season and see what it does. If the straggly bits annoy you you could trim off the longer ones.

It is Clematis cirrhosa, winter flowering but without the flower it is difficult to say which variety - it looks pretty healthy.

Thank you. We've been here for a couple of months, but there's been no sign of flowers. Do you think I should prune it & if so, when? I could try to tie it to the trellis higher up on the wall behind it where there's a rather tall rose. Would that encourage it to flower next winter?

The problem is most likely the container - it needs planting into the ground, with a support for it to climb up. That's what I recommend you do with it in March - either that or a much larger pot, 2 feet deep by 16 inches wide.

That looks suspiciously like the bane of my life - wild clematis or Old Man's Beard, but if that's the case, why would anyone in their right mind plant it in a tub? Definitely a clematis of some sort.

Thanks Bamboo & Gattina, I DID wonder if it was a weed!

I would put it in the ground with some fresh compost and tie it to a trellis. You can remove any untidy/unwanted growth. Keep it well watered and feed it occasionally and it should flower next winter.