As a JAR licence holder, the method of finding out requirements is no different whether you are talking about "A" or "H". LASORS and/or JAR-FCL 1 (no doubt you are familiar with FCL 2?
If your licence was issued by the CAA, then you cannot renew it in the States. The paperwork and money has to be sent to Gatwick. You use the term licence renewal. Renewal of course refers to a rating that has lapsed. The licence merely needs to be re-issued. JAA take very plain english and somehow use it to confuse, but its one of the easier piloting tasks!