Just be straight with them and tell them what you’d like to do. Have a rough idea of timescale and a decommissioning plan ready to explain. Have BCA cards with you to show you have cover, if that crops up.
To some extent your approach depends on what kind of private land it’s on and the background of who you need to ask (e.g. farmer, national trust administrator, private estate, large house with big land area attached, etc.).
Most land owners or managers just like to know what’s happening on their land. Imagine it was your house and you saw people in the back garden unexpectedly. You’d be concerned!
The other aspect to take into account is if the land is an SSSI (or whatever equivalent they have in Wales). In the Dales cavers have excellent co-operation with the body which administers SSSIs, largely due to a certain caver & forum member who performs that role for Natural England. Maybe the Welsh caving community has a similar contact person?
Hope the above helps.