In the future, if you encounter similar problems, you could consider buying things from the UK (or for that matter, from most other countries in Europe). I have had good luck ordering from country-specific Amazon stores.

Shipping is not negligible, but I find it is frequently surprisingly little compared to what I would have expected. A quick glance at the amazon.co.uk suggests that shipping an undercounter wine cooler to the US would set you back somewhere on the order of $85.

Amazon usually is good about shipping internationally for items that they sell themselves, but several third-party vendors on Amazon don't do so by default. I have sometimes contacted them directly, and discovered that in about 50% of cases, they are willing to negotiate special terms to ship to the US. Also, they should technically refund value-added-tax, when they do this. That's about a 20% discount, which can often make the extra cost of shipping more palatable.

The downside is that you won't have easy warranty service, and you also will have to deal with country-specific model differences. Dimensions could be a little different, but a quick check suggests that UK appliances are compatible in height with US models.

And of course, voltage is different. The UK uses 240V/50Hz, whereas the US uses 120V/60Hz. The difference in voltage is easy to fix. You simply ask your electrician to run a 240V circuit to the kitchen. That's possible in most households, although it might be difficult to do in some apartment buildings that get three-phase service.

The difference in frequency is usually not a problem. Modern electronics never care; resistive loads (i.e. heating elements) don't care; inductive loads (i.e. motors and transformers) might care. That's a chance you have to take. Or you could ask the manufacturer. Realistically, it is extremely unlikely that operating at 60Hz instead of 50Hz is going to cause problems. The reverse is more likely to be a problem.

This only leaves you with potential questions about building codes. There is nothing inherently wrong with European appliances. After all, hundreds of millions of people use them daily. But they might not have been tested by American test labs. And they probably won't even show up in American building codes, as they simply aren't a major consideration. This could be a problem when you get the kitchen inspected -- or it could be a non-issue. Depends on your local building department.