Who should apply for adverse possession? Us or buyer

Hello

We are currently selling our house and have a buyer however during this process it has come to light that the strip of land that is currently our side path does not legally belong to us. 
A housing development behind us bought the land in 1999 and it seems when fencing it off the boundary that was erected by them (which is currently our fence which encloses our side path) was submitted in the wrong place on the documents sent to the LR.

We have used that land for 20 years as our own, and the company who now own it have dissolved. We have enquired at length about new freeholders and all have confirmed they don't own the land or their boundary stops at our fence.

We have been advised that we need to now apply for adverse possession of the land to gain possessory title if granted. We have the evidence to supply to the LR and I'm aware of the process but I am unsure who should do this as we are in the middle of a sale.
Should this be us (the current owners)....and if so can the application be transferred to our buyers to look after if we are able to complete if they are happy? 
Or should this be our buyers that apply with our evidence, statement of truth etc?

To summarise, in order to keep our sale moving forward if all parties are happy, please can someone outline the process of who should apply.

Hello Sophie - anyone can apply so it's a matter for you/your seller to consider and confirm.
However the buyer really holds all the cards here as their conveyancer may say No, you need to apply. Or more likely, if they are buying with a mortgage, their lender won;t allow them to exchange/complete until the land is registered and included in the purchase.
So all down to the buyer, their conveyancer and/or their lender (in reverse order really) to decide.
If they say you need to then get your conveyancer to submit the application. Then ask them to ensure they request expedition to get things moving at our end.
The processing will still take time as a site visit will be needed and wider checks undertaken as appropriate. That time will most likely add up to at least a further 10-12 weeks of wait time as the two stages I mention must be completed and can't be hastened/expedited