I left the scheme for 9 months, 4 years ago and LV (Liverpool & Victoria) accepted the Motability no claims letter then.
I use an comparison site first and LV was a good bet for me so while had their price in front of me I gave them a quick phone call to see if they would accept the letter and she said they would. Right there and then I pressed the buy button and my private insurance was up and running. They never did come back and ask me for proof of no claims so assume its all on the insurance industries data base.
I can’t remember if I enquired about the need to inform them of adaptions, I might have done but don’t think I did but they had my driving license number and that states the adaptions I need to drive.