I've noticed that WotC and Paizo have slightly different design styles, as well as different ideas about balance. I'm actually not certain the two working together at this point is a good thing; their differences may actually translate to being worse products.
Really, it's an area where the two companies have evolved in different directions.
I don't think that's a good idea for Paizo. To me, it would basically be turning their back on their own fan base, which they've spend years and years building up goodwill with, to chase customers of another company's product.
And the whole point of creating Pathfinder was that they could control their own destiny, not rely on the good graces of Hasbro's legal department.
I'd like to see Paizo continue making the stuff they make now, but also making Dungeon and Dragon magazine in their subscription format. I dunno, maybe 20 bucks for each per month is too much? Paizo did a fantastic job with D&D stuff in the past, I would love to see them get to do more.
It seems like Pathfinder is popular but they've made most of the splatbooks they can make, right? Can they make a Pathfinder 2.0? What would that even be? It seems to me that it would be smart to do some 5e stuff. At the very least it's another revenue stream.
I think if we see some sort of OGL for 5E, there will probably be an adventure or two with dual stats for PF & 5E. Probably won't see anything in reverse from WotC except maybe the re-release of some reprinted 3E books or adventures possibly down the line.
I wouldn't expect them to become joined at the hip like they were back with Dungeon/Dragon, but many of Paizo's staff came from a background at WotC/TSR and you don't hear much enmity (if any) from them towards WotC. I could see them doing some joint projects, but they each have their own lines to maintain. I think they're both mature enough to understand they aren't in direct competition (those who are still playing PF only won't ever be persuaded to go to 5E, those in love with 5E exclusively won't ever switch to PF), but they do realize they have overlap with some of their fans.
I can't see that happening for a long time, if ever. Absolutely, if I had my druthers, I'd love to see Paizo publishing adventures or conversions for its adventures -- but why on earth would it do so? It has a very successful product line, the market leader (even with 5E out), and a history in which it was almost screwed out of business by tying itself to a WoTC license. Paizo wants people to continue to play Pathfinder, and very little business motivation to "help out 5E".
Perhaps one day, years down the track, if 5E completely swallows Pathfinder somehow, Paizo might be compelled to publish 5E adventures. But for now, ain't gonna happen.