Due diligence and 'buyer beware' should go without saying with any supplier you choose, but more so for a custom developed ecommerce solution supplier. Do your homework, has the supplier done work for other UK clients creating similar ecommerce sites, can you contact existing clients, etc.
Hopefully you will get a
client-supplier contract signed up before handing over any money (e.g. deposit). Remember that contracts are official legal documents which usually abide by the rules of a certain country, e.g. England and Wales. This is so that if things go wrong (and they do), you have a leg to stand on in the courts of say, England and Wales. When sourcing development overseas, which country do you select, and which country are both parties going to be willing to visit to resolve disputes? e.g. getting your deposit back because the supplier hasn't delivered.
Hosting will depend on whether the supplier's solution dictates their own hosting or not. Hosted managed ecommerce solutions are by their very nature only available on the supplier's hosting. Other solutions may also have certain hosting requirements, e.g. the supplier might only support the system if it is on their hosting platform. This is something you need to check. If you have a .co.uk domain name then there isn't much of an issue whether the hosting is UK or US, end even US hosted .com domains are less of an issue nowadays since you can inform the main search engines that the site is for UK via their webmaster tools/accounts.
Other potential issues:
1. An american design with americanisms, e.g. add to cart rather than add to basket
2. Some american web designs are more in your face, rather than UK designs which are more subtle, perhaps this is to do with heavy advertising in the states
3. Use of US stock photography, some sites have photographs of models that just look american!
4. Lack of understanding of the 'UK system' in terms of VAT, shipping (mainland, offshore, other countries), VAT on shipping, UK regulations for what gets displayed on UK company websites etc
5. Support, you come in the next morning at 9am, and there's something wrong with your ecommerce site, who are you gonna call? (it will be at least a 5 hour wait and time difference if your supplier is US East coast based and starts work at 9am)
6. Lack of understanding of the UK Payment Processors (e.g. Protx, WorldPay), or UK versions of US Payment processors (e.g. PayPal and Google checkout usually have different US versions to UK versions since some of their new functionality they just release in the US first)
7. Ultimately, do you know where they live and can you go round knocking on their door if issues escalate?