My view on this is the following:
First of all Richard Skinner is very reliable and trustworthy and I don't think he would intentionally lie.
Filling in the gaps...I would imagine that Richard Skinner thought he has Paul's home number, but in fact, it was his private work number he'd been given ("Here you are Richard, this is my personal number") - i.e. it would've been his personal manager's private number. He rang it and spoke to his manager. Paul's manager then got through to Paul - once he plugged his home phone in - and this was when Linda was taking the kids to school. This makes all the lines join the dots.
Paul then called Skinner at Radio One and had an emotional chat. If there was talk about a reunion, I would imagine it was something about him and John were possibly going to work with Ringo in January, and there are just some blurred lines in recollection, and perhaps Ringo has intended on working with John in December as some point. I think Paul and Ringo were communicating quite a bit around then, so they would've been quite aware of each others' intended activity.
Paul called Whistle Test later that evening too (a BBC TV prog.) but again did not want to be on air.
I think lots of things were mooted around that time, especially helping Ringo. We do know that John was planning on visiting the UK in 1981, so who knows what might've happened. Ringo would certainly have been the one person J&P would've come together for. Paul did keep quiet about things, and this is even shown in his Good Morning America interview a week or two before John's death, when he said as much about John (as in he has learned not to comment on the things John says). JL changed his mind all the time.
I am not sure how reliable Jack Douglas is, to be honest, as I don't know enough about him. This was the quote from Jack Douglas which could confirm the story which Skinner tells:
“He (John) and Paul planned to play on a Ringo album and that’s how they were planning to do it, and George (Harrison) had not come aboard yet”.
“George was already in a lot of hot water with John because of George releasing his autobiography and not really mentioning much of John in it. But I think they assumed that George would come along as soon as the thing got going.”
The album Ringo was about to record was going to be called “Can’t Fight Lightning“. Paul helped produce and record several songs and George also came on board. In New York City in November, Lennon gave Ringo the demos for “Nobody Told Me”. “Stepping Out” was another Lennon composition intended for Ringo to sing, as was “Life Begins at 40”. With Lennon producing, they set a date for 14 January 1981 to record the song(s).
Douglas also claims Lennon’s wife Yoko Ono actively blocked a Beatles reunion, saying “Let’s just put it this way – Yoko discouraged Paul coming around”.
“There was a writing session somewhere in the Dakota and there was one cancelled which John did not know about. He was told that Paul did not show. Paul was told that John was too busy to be working with him that day.”
The last para. seems very much unfounded in my view, although Paul was in NY for the Rockshow premiere in November but other than in Jan 1980, I don't think he was in the USA at any other time in 1980. It is possible that a Paul and John meeting to have talks could've been planned and cancelled though. As far as I am aware, the last time they spoke was on John's birthday 9 Oct '80.
I do think there are still many things which have not been told, sometimes people just never ask Paul! I am sure there are probably some more photos of John and Paul in the 70s taken by Linda which will one day see the light of day too.