Do not show your neighbour the snail before getting this achievement as you lose the snail once you have.
You need to go to the Town Centre (past the bus stop) and show the snail to the policeman, lady in the newsagents and the shopkeeper. Then return to your garden and show your neighbour. Once you have shown her the snail the achievement will unlock.
Note: I had already tried to use the snail on Gromit in the cellar, but it said there was no point when I tried, so I am unsure if you need to show Gromit, I would to be safe.