IP can* affect your rankings to a certain point, but unless there are serious issues with your IP, your rankings won't be affected.
IP DOES affect rankings depending on what IP you are using. As someone else mentioned, they also DO change from city to city, and even zipcodes. Google is very specialized like that. If you don't believe it, try out a few different proxies (go to starbucks and/or mcdonalds) and check your rankings after you've cleared all of your cookies. Chances are if you're in a big enough city, the results will not be the same. If you have more than one data center in your area as well, sometimes you'll see one set of results at one part of the day, and another set of results during a different part of the day. Where I'm located, there are two data centers nearby. One data center shows results during the first part of the day, and then late at night (around 10pm-12pm PST) it changes to the other data center and results are completely different. If you change your country preference in GWMT your website will be favored in THAT country (the one that you choose), so if you decide you want it to favor the UK, chances are your website might disappear or move way down from the US search results. It just depends on how strong your link authority and relevancy is to the particular keyword s that you're trying to rank for. I've tested this on multiple occasions, as much of my traffic is from international sources. I would highly recommend using backlinks from the country you are trying to rank in
instead of switching it on GWMT because then you have a better chance of ranking EVERYWHERE instead of just
one country like you would if you switched in in GWMT. Most keywords have people searching from multiple countries anyway, not just one. Take "Michael Jackson" for example. He's known internationally, so people search for that term not only in the US, but also in other english speaking countries like the UK, Australia, Canada, Ireland, France, ect. ect. Hope this helps.
P.S. Sometimes Starbucks and McDonalds use what's called "Wandering Wi-Fi", where they actually give you an ip that is NOT in your city. This explains different results to some extent, but if you check it and the ip is NOT located in your city, try a local library or coffee shop besides starbucks, as they are likely to have a local ip.

Hotels work too.