If you're getting a mortgage the lenders will have a big problem selling it as most lenders wouldn't be happy with the lack of formal easement, basically any problem which would hinder then selling it on easily in event of repossession.
Is the property registered?
I wouldn't be too sure about whether an indemnity policy would be issued it it's council land as I've had issues trying to insure on that basis previously. If the council are approached then any indemnity policy would be voided in any event.
With it being part of the garden and drive, it might be worth checking the deeds in case any mistake was made on the plans, I've seen it happen before and got the Land Registry (if registered) to sort it out. Get your solicitor to doublecheck the deed plans.
If the land is unregistered could there be a missing deed somewhere? Might be worth checking with the Land Registry esp if the issue affects other properties as a document could have been missed or not produced on first registration.
Any issues faced now would be faced by you if you wanted to sell, remortgage etc... so you could be making future problems for yourselves.
Do neighbours suffer the same problem? Could be a possible solution in there somewhere if so, I'm guessing your solicitor has a copy of the council's title so it might be obvious if it affects more than the property you are purchasing.
I wouldn't budge until a proper solution is found.
May be worth considering adverse possession due to the time frame, trying to get possessory title.
There is some info on registering prescriptive rights here...
http://www.landregistry.gov.uk/assets/library/ documents/lrpg052.pdf