I found a fantastic BBC Kent article which gives "gavver" as an example of an almost unique Gypsy Traveller word:
Romany Roots: Pookerin' Romany
A guide to language:
You may have noticed that words of Romany have been included in many of our broadcasts as well as in messages left on the boards. The language consists mainly of sanskrit words which are used in sentences using a standard English construction.
Sanskrit is of a form of language used in Northern India in the ninth century and therefore gives us a clue when the Gypsy people first left that area and began to travel the world.
In addition to the Asian words are others that originate in 'Cant' which was the language of the road used by the 'sturdy beggars' of Elizabethan England.
'Mort' meaning 'woman' and 'kenna' for a 'house' are examples of Cant words. Also mixed into modern day Romany Pookering are occasional words and expressions drawn from Cockney rhyming slang and other local dialects.
Some Romany words have also moved in the opposite direction and are used locally by non Travellers, these include 'cushti' meaning 'good', chavvie when used by Gypsies means 'son' or 'child' but is now used by many non Gypsies to mean 'mates'. 'Wonga' comes from the Romany 'Vonga' meaning money.
There are no standard spellings for Romany words because it is an oral language which has always existed outside the academic environment.
There are different opinions amongst Gypsy Travellers as to whether the language should be put onto websites or made more widely known by other means.
Some Romany words used by local Gypsy Travellers:
Atchin tanStopping Place
The paragraph I have emboldened is, I think, best suited to the question as it explains why the word has become part of the Kentish dialect and the paragraph below that tells us why Urban Dictionary has an article on gavver and gavva
Additionally, Manchester University have a brilliant dictionary project which details twenty-nine† words for policeman in the Anglo-Romani language along with a small amount of etymology.
gavva ['gævə] [from] garav- 'hide' (European Romani)
†Unfortunately this is probably down the the Gypsy Traveller community's reputation for violence and unlawfulness as well as police discrimination against the community.