I wouldn't use a climber at all. Had you thought about planting a fresh hedge in front of it? You would get a much neater effect than climbers. Or you could plant a mixture of shrubs. What would do well there would depend on which direction the hedge faces and what kind of soil you have, but whatever you plant the ground will need a lot of improving because of impoverishment by the conifers. The trouble with any rapid growing shrub or climber is that they don't normally stop at the height you want - usually the fastest growing ones will be the biggest.
Shrubs could be planted further away from the boundary than climbers too so it would be easier to dig the planting holes - the ground immediately under the conifers will be hard, dry and full of roots.
It would take a few years for shrubs to get tall enough to hide the bare branches but they would attract attention away from them in the meantime. YOu might even plant a small tree in the worst place if you have room.
A bit more info would help us to suggest suitable ones

I think I'd put a 6ft fence up in front of it. You're not going to get anything to grow under that awful leylandii as Steragram says, what with all the roots and dry nutrient stripped soil. That would hide a lot of it.

Good idea.

Hi, I was interested in this question as we have one of these which was topped and stands about 20ft with bare sides where it was cut back - I gained at least 6 ft of garden when they were cut!
I decided that as it wouldn't grow again it would be left as wind protection and I would grow climbers for the birds to nest in. I left the ivy, all the cuttings of the trees were woven into and through the stumps - there is at least a 40 ft run of this hedge, so lots of it! I planted anything and everything climbing to try to hide it and (depends on aspect of course) the most successful was Solanum crispum Glasneyin which at the moment is a mass of purple/blue flowers, (https://www.rhs.org.uk/Search?query=solanum+crispum) otherwise one of the Virginia creeper types or Goldheart ivy which is tough as old boots. There are loads of ivies which are pretty and will cover it. Otherwise I would suggest a row of alternate Laburnum and the black lacy elder which has lovely pink flowers, both of which can be cut back hard.
Buddleija or Clematis?
Best of luck, think of it as a vertical garden rather than a problem!

Just realised you are in Norfolk, as am I, it must be a trait here to inflict Leylandii onto your neighbours!

Beware! I had a leylandii hedge, 10ft from the house , planted by the previous owners. I always kept it neat and tidy, paying to have it cut twice a year. Over the years it looked very smart but, of course, grew taller. Two years ago I had some subsidence to the integral garage wall and the insurers decided it was caused by the hedge, so very sadly, I had to have it removed and have replaced it with a fence. Needless to say, although the cause of subsidence has been removed, my buildings insurance has rocketed!

Nasty horrible things Leylandii, because they are not kept under control most of the time. They should be banned as far as I'm concerned (as I have said many times)!! I don't think anyone knows how big they can get but I've seen some whoppers in my time.

Many thanks to all that took the trouble to answer my enquiry.This has been my first use of 'growsonyou' and I am confident it wont be the last.

Oh, welcome to GOY.