The only JAA country that certifies foreign flight schools is the UK, and several school in the US have obtained this certification. So you can do a full JAR-FCL PPL there, send in the paperwork to the CAA at Gatwick and receive a license from them.
As this license is JAR-FCL compliant, you don't need to exchange it for a Belgium-issued JAR-FCL compliant PPL. In fact, you can only trade it in for a Belgium-issued PPL if you live in Belgium for more than six months each year, I think.
Any JAR-FCL PPL is good to fly on any aircraft registered in any JAA state. So with your UK-issued license you can fly G-reg, D-reg, PH-reg, OO-reg, F-reg and so forth. For non-JAA registered aircraft there's usually a not-too-complicated process to get your license validated, but this differs from country to country.
Once you have the license, where do you intend to fly? Just on European holidays or also at the place you live?