Title plan shows incorrect boundary, two titles overlapping

My title plan overlaps with my neighbour's. The complication is that the neighbour is the owner of the lakebed of Windermere, and their title plan asserts that the lakebed that they own extends underneath my land.  I believe this to be a drafting error on the part of the predecessor Council's surveyor when the first registration took place in the last twenty years. Their boundary is defined as the lakeshore at the OS Datum at a certain height above sea level in the 1930s. It's inconceivable that the natural lakeshore executed a variety of organised right angle and 120 degree bends as their plan is drawn.

I have written to the Council's lawyers suggesting where their boundary ought to lie, and asking if we can reach a mutually agreed position on this, but beyond a polite response after five months have had no response from their estates team.  I can not get on with repairing my small slipway until we have this resolved.

My property is the area behind the green line. The green line area rises away from the lake.  The Council's deed shows that their property is everything on the near side of the red line on the attached photograph.  The strange angled shape described by the red line can not possibly ever have been natural lakeshore.  I am proposing we agree that the boundary goes from the front of the boathouse on the left, to the front of the one on the right along to point "F", and then back to point "G". The Council has a vested interest in delaying this since they can change "encroachment rent" on any structure built over the lake on their lakebed.  But the slipway is manifestly not on the lakebed, and never was.

I now wish to apply to have their boundary corrected.  What do I do next please?

Adrian - make an application using form AP1 and supporting documents as appropriate - Change the register (AP1) - GOV.UK

Thank you Adam. Is that the case even if the correction is to be made to a title which is not mine?  i.e. I'd be applying to change someone else's (the Council's) title.

Adrian - you would make your application as appropriate. If you are uncertain as to the supporting evidence/the scenario you have then I would recommend using our contact form to submit the request to Amend and the support team can check and respond as appropriate - Public guidance: dealing with land and property - GOV.UK

Will do, thanks.