I have a small potted lime tree which I have keep in a conservatory, it has developed a sticky residue on the leaves and branch joints. Please could you acvise?

check for scale insects and aphids and if you find them they will need to be killed/removed~ try provado vine weevil killer. In order to make the plant look better wash the leaves with a sponge with slightly soapy water(very small squeeze of washing up liquid)
I have calamondin oranges and a cross lemon with grapefuit which also suffer from caterpillars which love the leaves!
It might be a good idea to give a foliar feed of winter food as well.

Sounds like scale insects to me too. Check under the leaves for small flat grey insects stusck to the leaves. They will be sucking the sap from the plants and will eventually kill them. They can be removed with methylated spirit on a cotton bud and, once removed, the plant will flourish almost immediately. Good luck!