Is it possible to recycle an alkaline battery?
Yes, in theory. Everyready has 4% recycled materiel in some of its batteries.
The application I am thinking about is that in 3rd world countries often access to western markets is limited. You can't just pop into a store and buy batteries. Would it be possible for a small factory to be in these places and the locals could bring their batteries in for recycling?
No. The chemistry and processing involved would be far to complicated for some sort of rudimentary factory. There is also the economic factor. Even with a well educated work force such a factory would be very expensive to run and need a massive number of used batteries to recycle.
Part of the expense that recycling would offset is the cost of disposing of the used batteries. For example NiCd batteries in a landfill are bad news because of the toxicity of cadmium. Third world countries generally have much poorer environmental regulations than the developed countries.
To answer the additional question...
The reactions for an alkaline battery are the conversion of zinc to zinc oxide and MnO2 to Mn2O3. So, to recycle a battery of this type you would have to reverse these reactions. Is that easy to do?
No it isn't easy. You can't just mechanically separate the zinc component of an used battery from the manganese component. You'd have do something like dissolve the mess with acid then separate and process the two elements.