Of course, there are Josephs and Julies and Barries and A(l)lisons on both sides of the Atlantic. What differentiates them is their nicknames. I've known a few Allisons in the US, but none is regularly called Alli, but here, where it's usually spelt Alison, most are known to at least part of their social circle as Ali.
Both AmE and BrE use -y (or -ie) as a diminutive and marker of affection, as in Jenny or Maggy. But BrE (and some other Es) also make a lot of use of clipping (i.e. shortening) a name and adding -s. Some examples:
|Julie/Julia||Jools (or Jules)|
In AmE, the Friends character Phoebe was called Phoebs, but other than that I can only think of (the rather old-fashioned) Babs for Barbara. I can't help but see Madonna's UK nickname (spread by now to the US), Madge, as being related to the phenomenon. After a voiced consonant, the -s is pronounced [z], and it's a short jump from [mædz] to [mædʒ]. Jos is another common BrE clipping, but in this case the s (pronounced as unvoiced [s]) is not added but retained from Joseph. I'd never heard Jos till I met two here, both now 15. One now opts for the 'cooler' Joe.
My old university in South Africa, University of the Witwatersrand, is commonly known as Wits, which led many of my American correspondents, unaware of the diminutive -s, to address my mail (BrE prefers post) to "University of the Witswatersrand".
Another common personal nickname, via a different history, is Bazza or Baz for Barry. One also hears Shazza for Sharon and Mozza for Maurice or Morrissey, etc. (Click the link at the start of the paragraph for more examples.) Tabloid newspapers seem to like to dub people with -zza names, for some reason, but I do know of an unfamous Baz(za), a Shazza, and a Mozza, though the names are only used in very informal settings. In Bridget Jones' Diary, the character Sharon is nicknamed Shazzer, which is pronounced like Shazza.
[This paragraph added 10 July:] While some of the tabloid names make a -zza out of sibilant sounds--e.g. Gascoigne-->Gazza, Prescott-->Prezza, what's interesting to me here is how the -zza ending is added to the first syllable of a name whose second syllable starts with an /r/. I'm investigating this--but if you know anything about it, leave a comment!]
On the other hand, there are lots of American nicknames that are foreign here, including Bud(dy), Chip, Trip, Muffy, Buffy. Not that these are very common in the States, but they are really American.
Postscript (12 July): When I posted this, Better Half said "But those aren't nicknames." In my reading today I discovered the better term for reduced 'pet names' based on a person's given name: hypocoristics. So I guess BH is right. Don't tell him--it'd upset the whole balance of know-it-allness in our house.