What is a 'limey' and does the word have a negative meaning?

A name for the British. Not necessarily negative.

https://en.wikipedia.org/wiki/Alternative_words_for_British#Limey

Derived from the naval use of lime juice to combat scurvy. Some Americans think is has negative associations and try to use it as an insult. Over here, we couldn't care less. However, according to the Royal Navy, lime juice wasn't as good as the original issue of lenom juice (link no longer available) so it would appear that the USA was way behind on this one. The lime juice, which was cheaper, was brought in in 1875 but only contained half the quantity of ascorbic acid. Obviously the issue of antiscorbutic juice was only noticed when the lower quality stuff was replacing the original issue. Interesting. I didn't know that one before.

Long before vitamins were recognized as being essential to the diet, medical officers in the British navy noticed that even extreme cases of scurvy cleared up rapidly if the sufferer ate citrus fruits. The Royal Navy therefore instituted the practice of carrying stocks of limes and giving each sailor a daily ration of lime-juice. This resulted in British ships being called lime-juicers by U.S. sailors. Over time, the scope of the term was extended to include the British sailors and, eventually, all Britons.

Just to put your mind at rest, Limey is no longer considered offensive but, originally, it was intended as an insult.

http://www.takeourword.com/Issue067.html