Ownership of the bank/hedge - has this been determined? Is it yours or theirs? If it's definitely theirs (regardless of who "they" is) then, if they haven't got their act together by the time you want to get cracking, get a pet solicitor to write to both of them giving 7 working days for them to answer.
My best is that it will be the LA, not the HA. The HA only really "owns" the BIG roads, such as certain dual carriageways and the motorways.
2 - You cannot remove the hedge/bank without getting permission. It's a basic tenet of the Streetworks act that you MUST have written permission and the appropriate training/insurances before excavating on a public highway.
This is done for a good reason - namely, to prevent any bugger who gets the notion from digging up the path or road wherever they like. Which takes us on to...
3 - Unless you're an accredited contractor with around 5 million quids worth of insurance, they wouldn't even allow you to put it back. They will send round one of their approved contractors, or their own DLO lads, who will then stretch the job out to two days, and spend their time supping tea and struggling with the Daily Sport crossword, all at your expense.
From past experience I can more or less guarantee that this is a Local Highways Department issue, and you will need their consent before removing the hedge/bank. You will also need their agreement to the creation of a new access point onto an existing public highway, and then you will need even more permission to form a crossing between the road itself and your planned driveway.
So, it's best not to piss them off too early! Jolly them along, with regular phone calls asking for progress updates and get them to give you a 'D-Day', a date in writing by when they will have sorted it all out, one way or t'other. Be polite but persistent, get them to arrange a site meeting so they can swan around in their dayglo jackets with cans of yellow spray paint while you ply them with tea and Chocky HobNobs.
Eventually, you'll get your own way, but you have to let them feel as though they've "granted you permission" rather than caved in uner relentless mithering. It's a power thing that is endemic in most LA Highways departments! ;)