ivierre,
If you have 20 hours of JAR instruction, then it would make sense to carry along that route towards a JAR PPL and that can be done two ways:
1) stay in blighty and finish your training here, or
2) go to North America and attend one of the seven schools on the link provided by vincent.
Any school, regardless of where they are situated will want to see what you have been taught so far, obviously.
If you take the JAR route through to a succesful conclusion - you get a JAR licence - there is no conversion required at all.
You will know this already but it's worth reiterating - the school you choose do not provide you with your licence - the CAA does that.
PD