Tourists from the USA in the UK always sounds funny when pronouncing place names that have "ham" in them. The UK English way of saying these place names is Buckingum and Birmingum, not BuckingHam and BirmingHam.
I would have said BuckingAm myself.
Speaking of the UK, how on earth did Worcester come to be pronounced "Wister". At least that's the way it's pronounced in the U.S. northeast, when referring to the city in central Massachusetts. The Brits threw New England a whole bunch of curveballs.
I'd pronounce that Wooster
Actually I always hear it pronounced Wuhster. Although maybe you did mean "oo" as in "wood" (the sound I hear around here) rather than "root".
Yeah it's a schwa, rather than an oo.
Likewise, Wooster as in Bertie. On that note, I've always found the pronunciation of Towcester amusing. It's a town... and it cooks bread!
Like the pronunciation for "forecastle" (fowk-sul), it's a pronunctiation borne out of expediency or laziness.
And don't forget Loogerbarooger!
(Loughborough)
Brits visiting Alabama would be equally wrong to omit the ham from Birmingham.