I would like to know if it is possible to apply for a new UK visitors visa in your home country (Nigeria) two weeks before old one expires. The reason is because I have to be back in the Uk in the first week of January and since my visa expires on December 19th 2014, the application process which takes about 3weeks extends into this time.
Of course. A visa is merely a statement of period of validity - you're asking the country for permission to enter between dates x and y.
If you also need permission for dates y to z, or different dates entirely, you can do so.
Depending on the country, reason for second visa, and type of visa, they may either allow you to have two valid visas in your passport (I had this with Tajikistan at one point - two 'windows' during which I could visit) or would invalide the first one. Since you'll be at home, either result would work for you anyway.