Funnily enough, I just spoke to a chap that runs an advertising trailer company.
According to him, and don't take this as gospel, they are subject to the same planning permission laws as normal advertising boarding. The fact they are on trailers is irrelevant.
However, having a lot of experience in planning permission cases, I know that you can apply for planning retrospectively and it can take months for that to come to a decision. And, you can appeal the decision, which can also take months.
On the marketing side, you want a short sharp notice and Call to Action on a trailer - people don't have much time to write things down at 60mph. ;-)
Some people put on shorter urls than their main websites so it's easier to remember. E.g. debtexpert.co.uk rather than anotherdebtcompanywithalongname.com