Sorry in advance if i repeat anything already stated in this thread but i've only briefly skimmed over it. Didn't want to influence myself too much with others opinions until i'd given my own interpretation of what happened to Cordy.
First of all, this was clearly a longterm plan of Jasmine's. We discovered in Inside Out that she had been manipulating events for a very long time, certainly a lot of what had happened to Cordy since season one of Angel had been influenced by Jasmine's wishes. This isn't to say that Cordy didn't make her own choices though, she and the other characters all had their free will to choose their own path. All Jasmine could do was create events that would influence them to make the choices she needed them to make. Similar to the way Cordy made the slot machine pay out in The House Always Wins in order to get Angel where he needed to be.
But essentially Cordy, not to mention Angel, Darla, Wesley, Connor and the others, were all being positioned like chess pieces on a board, to bring about Jasmine's birth.
One such instance was Cordy being given the visions, something no ordinary human could ever survive for long, which eventually forced her to be made part demon. Now is it just me or did anyone else notice that the powers she displayed throughout the end of season three were very similar to what we eventually saw Jasmine do. The calming influence that her "light" brought about when Connor was about to attack her in Benedicton looked to me to be exactly what Jasmine did with her own powers when her very presence would remove all hatred and violence from the onlooker. It is clear that the alterations made to Cordy were a lot more far reaching than just to allow her to survive the visions. She now had the correct demonic makeup that would make her capable of giving birth to Jasmine's physical form.
All that mattered now was to get Cordelia to the higher planes so that Jasmine could infuse herself into Cordy. It was during her time as a higher power, between Tomorrow and The House Always Wins, that Jasmine possessed her. that was the whole reason for her being taken up there in the first place.
The amnesia was just a totally unexpected side effect that even Jasmine was not prepared for. Returning to earth and her physical form caused Cordy to forget who she was and where she had been whilst Jasmine, for some unknown reason, became dormant. This was until Lorne's spell reactivated Cordy's memory and woke Jasmine up.
However this was not the point where Jasmine took Cordy's place entirely. In fact there is no sure way to determine exactly where Cordy was completely submerged, although it's fair to say that by the end of Calvary it was entirely Jasmine running the show. Lilah's murder showed that much. After that it was very clearly not Cordy making the decisions.
However between Apocalypse Nowish and Soulless it is much more uncertain. Up until Long Day's Journey t is obvious that Cordy isn't totally herself and the decisions she makes, sleeping with Connor being the obvious example, are of jasmine's doing. Even so we see many small bits of evidence that show Cordy is still present. The nightmare about the Beast she had at the start of Apocalypse Nowish, the "morning after" look she had on her face when she woke up in bed next to Connor, the intense regret she showed when it became clear that Angel knew what had happened. All those things were not done in front of anybody so there was no reason for Jasmine to be acting, clearly it was still Cordy in there.
However Jasmine was able to take Cordy over for brief periods without her realising, the various murders she commited, including Manny in Long Day's Journey, and the stealing of Angel's soul, were only possible because of that.
There is still evidence that Cordy was around during her conversations with Angelus in Soulless although by that point in has become very hard to tell how much of what was being said was her and how much was Jasmine, certainly nothing that Cordy does in that episode could not be considered a part of Jasmine's manipulations. By the end of Calvary though, as i previously said, Cordy was no longer in any control of her body.
So to summarise, Cordy was altered to be Jasmine's vessel in Birthday, she was only actually possessed though during her time as a higher being. Jasmine was awakened in Spin the Bottle but took control very gradually and had only taken over Cordy completely by the end of Calvary.
Wow that was longer than i intended! 😉