Right, since you don't want to be an in-house lawyer, you won't need to qualify as a practitioner in any particular jurisdiction (although being a qualified barrister/advocate never does any harm when job hunting!) So it won't matter where you did your degree.
The problem you face is more likely to be that of finding a route into these international organisations. You'll have to try making contacts while at University, and probably find a way to do an internship. This can be very hard. They don't recruit in large numbers and often, for policy/research work, they hire PhDs or take secondments of senior people from national government.
Your best bet might be to get into a UK-based organisation that does economic development work (Oxfam, ActionAid) and use that to build the right reputation in the right circles. But even getting into these organisations on the policy/research/parliamentary/campaigns tickets is very tricky and competitive. It makes becoming a solicitor look like childsplay. You will probably have to put up with being a junior staffer on a very low salary at the start. But the work can be really fabulous.
PS Lewis is once again talking freestyle. The best thing of all if you want to work for these organisations is to be a highly qualified (PhD) person from a country with economic development problems.