I agree with some of the comments the Dispatchers role is fast disappearing. At BA they will soon be titled as Turnaround Co-ordinators (TC), this covers two issues.
So much more of their work is performed by the centralised load control planning system. This in effect means that flight load sheets or a LIRF (Load Infomation Report Form) can be generated automatically from a computer at anytime and in effect at any station you operate too. For example the same a/c types with near enough same load/ pax figure etc can be duplicated year in year out.
Within the UK the CAA has directed that the TC will now be responsible for all the activities conducted around the a/c and will manage them accordingly. This is very much seen as a health and safety role as previous events have shown that no-one around the a/c ramp operations knows who is in charge which is a H & S failure.
Sorry to go on but this may mean that the role may actully increase. If you wanted to go into it with BA you would probnably have to work first as a Customer Service Agent or a ramp agent as Dispatch is promoted from within and this is the case with many airlines and ground handling companies.
Good Luck