As a British person, I must register my disagreement. My pronunciation is much closer to the second alternative.
There may be some confusion (perhaps even on the part of Wiki editors) because of the name of the county, Lancashire, of which Lancaster is the county town. “Lancashire” is pronounced “LANK-uh-sheer.”
Serious question: what has caused the erosion of in particular the much older francophone and Spanish names? One would imagine when these places were named they approximated the original pronunciation - yet there are so many mismatches, are we to assume that English speakers simply overwhelmed the original populations and pronounced it as it was spelled? Or is there something else going on?
Another vote for “LAN-CAS-ter” as a Brit.
I’m a Brit and I say it more like “LANG-kuh-stuh”, but then I’m a southerner. I believe the locals pronounce it as jjimm says.
Well, I am a southerner. (Admittedly, I did live in Yorkshire for quite a few years, but you can’t believe anything they say about Lancashire.)
As I said, the confusion probably arises from the county name, which is probably heard a lot more often than the name of the city. Lancaster is, now, a relative small town in Lancashire, one of Britain’s most populous, and historically significant counties.
And Featheringstoneshaugh is pronounced “FAN-shaw” which is just cheating at Scrabble. 
Mississippi has a couple of strange ones - Saucier being the primary example. I always want to pronounce it SAW-see-er, but it’s pronounced sew-SHAY. Kosciusko is another: it’s a Polish name so it should be pronounced Ko-SHOE-sko. People here pronounce it kos-ee-ESS-ko. It does aggravate me when I hear a newscaster pronounce Biloxi buy-LOX-ee. I don’t even think you can buy lox down here.
My husband got a kick out of me back when we were dating and he was driving around town: I asked him what a pas-SKAG-you-la was. (Pascagoula)
It’s spelled Featherstonehaugh. And you can’t use it in Scrabble as it’s a proper noun.
The Hialeah case is a sort of phonetic tennis game between three languages.
Montpelier hurts my ears.
Beat me to it. I did a double-take the first time I heard it pronounced.
We’ve got the town of Berlin, CT and it also sounds weird because the accent is on the first syllable (i.e. BUR-lin).
Same story here in Connecticut.
Deh Moyn
Not technically a town/city name, but I grew up in Chicago near Devon Ave, which is pronounced “de VAHN”. It was years before I found out that the county in England was pronounced “DE ven”.
Delhi, Ohio – Dell-High
These are based on standard British pronunciations. I think.
Kind of. Lancaster has been discussed several times upthread. I’m not aware of any places called Beaufort in the UK, but BEW-furt would certainly be a plausible pronunciation, by analogy with Beaulieu (BEW-lee). The name Beauchamp, however, is generally pronounced “Beecham”, both as a surname and in place names such as Shepton Beauchamp.
It may be more regionalization than Americanization that can be blamed for some that I’ve found interesting:
Pee-roo for Peru
Kay-roh for Cairo
Bah-li-ver for Bolivar
Jur-dun for Jordan
May-run for Marion
Kay-diz for Cadiz
Bew-furt for Beaufort
Mye-lun for Milan
Muffs-burr for Murfreesboro
Huh-lee-nuh for Helena
Mur-ree for Maury
Sheb-vull for Shelbyville
Isn’t this pretty close to the original English pronunciation?
Coeur d’Alene, Idaho - kor-da-LEEN
We have more Indian names than foreign ones, but my brother lives in Moscow ID(moss cou) and there’s a Madras (mad-drass, almost sounds like mattress) OR. Then there’s Oregon, Ohio which is said so wrong that I’m not even going to say how they pronounce it. OK yes I am. It’s not Ore E Gone. It may be your town but there’s more Oregonians so say it right!
I’ve only heard Beaufort said like Bew-furt, I didn’t know it was said another way.
And if you go to Newark, Delaware, don’t call it NOO-urk (or NYOO-urk) – that’s the city in New Jersey. It’s “Noo Ark”.
In Kansas there is Marais de Cygnes, pronounced MARE-uh-du-ZEEN