Here's one possibility...during World War I, the cartoonist Bruce Bairnsfather created a popular character called 'Old Bill'. He was a grumbling veteran soldier with a large moustache. In the 1930s - perhaps because many ex-soldiers had become policemen - the name became attached to the Metropolitan Police.
It has to be said that the Metropolitan Police themselves have a website - click
here - that offers a whole raft of other possible explanations. If
they don't know for sure...who does?
If you click
here the link will take you to the website of the noted etymologist and lexicographer, Michael Quinion. His view on �Old Bill' certainly seems to be that Bairnsfather's cartoon character is the original source, as he does not even give house-room to any of the other supposed possibilities.