Taken from Wikipedia
Limey
The term is thought to have originated in the 1850s as "lime-juicer",and was later shortened to "limey".It was originally used as a derogatory word for sailors in the Royal Navy, because of the Royal Navy's practice since the beginning of the 19th century of adding lemon juice or lime juice to the sailors' daily ration of watered-down rum (known as grog), in order to prevent scurvy.
Eventually the term lost its naval connection and was used about British people in general. In the 1880s, it was used to refer to British immigrants in Australia, New Zealand and South Africa. Although the term may have been used earlier in the U.S. Navy as a slang word for a British sailor or a British warship, such usage isn't documented until 1918. By 1925, its usage in American English had been extended to mean any Briton, and the expression was so commonly known that it was used in American newspaper headlines.
Pommy or Pom
The term pommy, pom or pommie, in Australia, New Zealand and South Africa, usually denotes a person of British heritage or origin. Less commonly, it can refer to anyone from Britain or the United Kingdom. A non-derogatory term, it was ruled no longer offensive in 2006 by the Australian Advertising Standards Board and in 2010 by the New Zealand Broadcasting Standards Authority. British people or those of British origin consider the expression neither offensive nor racist when used by people not of British origin to describe English or British people and acceptable when used within that community. But the community group British People Against Racial Discrimination was among those who complained to the Advertising Standards Board about five advertisements poking fun at "Poms", prompting the 2006 decision.
There are several folk etymologies for "Pom," some of which are false etymologies. For example, there are rumors that the word's etymology is related to prisoners, such as "Prisoner of Millbank," but this claim is suspect. A more likely theory is that pommy originated as a contraction of "pomegranate".According to this explanation, "pomegranate" is extinct Australian rhyming slang for immigrant. A popular alternative explanation for the theory that pommy is a contraction of "pomegranate", relates to the purported frequency of sunburn among British people in Australia, turning their fair skin the colour of pomegranates. However, there is no hard evidence for the theory regarding sunburn. What many British emigrants find a source of irritation is when Australians refer to all Britons as 'English.' The United Kingdom comprises Scotland, England, Wales and Northern Ireland; therefore Poms can be from anywhere in the UK, and not just England.