A good product for the purpose of killing stumps is a product called “Dax Root-out” which contains ammonium sulpamate.
This product can be safely used around more prized planting providing it is applied to just the cut stump of the plant to be removed.
It has the added benefit of actually adding nitrogen to your soil rather than leaving a nasty residue.
The Ideal time for application is during May-June and directly after the cutting back the climber, but I have had good results at other times of the year.
Drill a slightly slanting hole approx 2.5cm (1 inch) diameter as deep as is possible into the top of the stump, that is if the stump is wide enough.
Also make downward axe slashes into the bark on the sides of the stump approx 2.5cm (1 inch) deep if possible.
Next pack the “Dax Root-out crystals into these wounds.
Finally cover the stump with a clear plastic cover to prevent dilution of the crystals with rainwater, leave this in place for 1 month.
If you cannot find “Dax Root-out” in the garden centres you can also use “SBK Brushwood Killer” or “Round-up” although these are in liquid form and must be painted onto the plants open wounds.
Under no circumstance employ the old method of applying diesel as a stump killer, this will definitely leave a residue in the soil to affect plantings.
A stump killing product will take approx 7 weeks to totally kill the plant, but for the stump to decompose completely can take in excess of 3 years.
Of course it may be possible to dig the stump out after a year.