What to plant by the fence (climber?)

Wait until the neighbour's out ( you don't want to distress them unnecessarily), stand on something (stable) so you can see exactly where the vertical posts are on the neighbour's side. Mark the position on your side, drill or screw vine eyes in if the posts are wood. If they're concrete you'll need to drill and use rawl plugs. Stretch wire between vine eyes.
Must admit I've never bothered with tensioners, but to do a professional job that's what is recommended, so the wires don't sag. If you're able to do all this discreetly when they're not around, they'll never know......

 

How long is the fence too - what size of area is there for two large climbers?
I believe most jasmines become pretty big, and roses will need decent soil and moisture, so you would need to create a proper border with suitable soil for them to get them established well, and then train them well to get good coverage.
You'll get advice from those who grow them though - I don't like roses, and the star jasmine doesn't survive where I am, so I can't advise.

 

I don’t think the neighbour would bother, my worry is more about making sure the wiring is firmly attached to the fence as maybe the drill doesn’t fully get into the post

the fence is quite big, probably 6-7 meters wide, it’s not too tall (2m maybe?)

The soil in Milton Keynes is crap, it’s pure clay, so I was thinking on digging quite deep and fill it with John innes 3, don’t know if that sound reasonable? I would prefer it rather than buiding a border

 

Does the fence have concrete posts ?If so and you don't want to drill into them have a look at the wire anchors Rivelinglen sell . They also do trellis kits.

 

No - don't take the soil out. Add loads of organic matter - and I mean loads. That's by far the best way to improve the structure, and clay is the best growing medium for most plants, as long as the drainage is good. Adding that goodness also means it can retain moisture better in long dry spells.

You need to make sure the plants have a good space to grow in - the grass is going to compete for water and nutrients, which can't be underestimated.
Make sure when you plant, you don't plant too close to the fence either. It can be a rain shadow.

 

do you mean adding it on top of the existing soil, or removing a couple of inches of the existing soil and add john innes there?

Also, i heard that jasmine and clay are not good friend because of drainage...