Claiming possession of a part of my garden leased from the council?

I’d suggest it was very doubtful you’d succeed in an adverse possession claim – the council is likely to have records of the original lease & could prove their ownership.
Adverse possession laws were overhauled recently to avoid this type of land sitting.
Additionally, judges don’t look kindly on this kind of action.

I thought adverse possession was only possible on land where the owner wasn’t known.

If an adverse possession was granted you would only get processionary title with the owner able to make a claim on the land until you were granted absolute title, a further 12 years on. you’d then need some sort of indemnity insurance to protect against that happening–which I don’t think you’d get as you know the land belongs to the council.
Personally I’d contact the council and ask to buy the land –saves worry about issues with title later. Also you can say that you want the last 20years of goodwill, where you have maintained the land for free to be taken into account when negotiating the sale

Would the local council want to expend the powder and shot over this issue?
They are all short of cash?
Give them £500 and tell them they are lucky.

You may have a claim, but I suggest you look at a case called Pye v Graham. It went to the House of Lords, admittedly on a laregr area than a garden.
It is possible to make a claim over registered land.
One word of warning, is that if you raise the issue with the Council, then your ability to claim will be diminshed – you have to act as though you are the owner, occupying without permission and without payment.
If the land is of sufficient value to your property, ythen I suggest you get a good local property lawyer involved – but speak to them before you contact the Council.

the Govt published this document in September this year, which seems relevant: https://www.gov.uk/government/publications/adverse-possession-of-registered-land/practice-guide-4-adverse-possession-of-registered-land

Reply to the comment left by Chris Bradley at 25/11/2020 – 10:19
Am I right in assuming you mean possessory title rather than processionary?

Hi. I have personal experience of Adverse Possession claims, be it a long time ago so may be rusty on the current legislation. My recollection is that you cannot claim adverse possession if at anytime during the 12 year period you were a tenant or had communication with the land owner about renting the land. If you can demonstrate 12 continuous years without interruption of opening maintaining and having possession of the land I think you have a very strong claim to acquire it. It doesn’t matter if it was council or privately owned. Best to get a Solicitor or put in the claim to land registry yourself. It’s not so difficult and see what response they give if any at all. Goodluck.

The law changed in October 2003 so your case will be stronger if you were there before that, you say 20 years so this may be the case