The situation is much more complex than you make out.
First off, American English and British English are not dialects, but rather groups of (often starkly different) dialects of English. Both sets do have prestige dialects. Most likely if one looks into it, the sets of vowel sounds used in the prestige dialects of RP and Standard American English will be a bit different, and one may have more in total than the other.
However, this doesn't really tell you much of anything. SAE is not a dialect that broke off and changed from a forever static RP dialect. Rather, the various American dialects evolved from the various dialects spoken by the settlers who came over from England, few of which would have spoken the "prestige" dialect of English of their day. RP itself appears to have evolved from the speech patterns used at Oxford, and probably has very little to do with how much of anyone else in England spoke in the colonial period. In fact, many speech patterns that modern Brits view as "Americanisims" are in actuality old features of the language which RP has discarded, but some American dialects have not.