I had the feeling they were resting on the inherited wealth of their empire for rather long now. They always wanted special treatments in the EU. They even pay mad bucks on their royals for basically nothing.
Maybe this Brexit will go down bad and they will change their ways and join the EU in 10-20 years as 'real' members.
But I don't have many hopes in that. Probably they will end up like Switzerland or something :/
The first being that the Royal family actually makes the UK money because they cost less to maintain than what is made up for in brand image and tourism benefits.
The second is that because they don't pay tax, it means their enterprises are not as efficient and the return on the crown lands is about half of what could be expected on a normal enterprise.
The source for both these viewpoints are videos on YouTube. The first is from CGPGrey (I think), while the second is a reply to that video.