I think it could work. It may not be something that people naturally gravitate towards, but if you call her Wren, everyone else will catch on.
I feel like I'm pretty liberal with nicknames, and I confess I don't hear or "feel" it enough. I agree that if you call a girl Wren all the time, people will call her that, and there are certainly people out there with nicknames, like Bear, that don't go at all with their given names, like Leanne. But I wouldn't personally link Wren with Serena... Sera, Rena, Sea (prn like see-uh or see, either way) all feel more natural.
It's too much of a stretch. Firstly, there's no 'w' (bug bear of mine: Lauren and Maren etc should be Ren not Wren, what's the point in adding a silent letter? Natalies are never Gnats or Knats :p) and secondly there's no 'ren' sound. Serena is 'reen'.
The Welsh name Seren would work though! Seren nicknamed Ren.
I love both Wren and Serena! Honestly, if you love it, I think it could work. A couple years ago I tried to do Serena nn Wren, but I got lots and lots of comments about how it was a huge stretch. But if Evelyn (EV-eh-lin) can get Evie (EE-vee), then why can't Serena get Wren? As for Wren vs. Ren, I think you could easily opt for the "W" spelling. The association to the bird is much better than the association to a gnat. :) Plus, Ren has always looked harsh and quite masculine to me (maybe thanks to Ren and Stimpy? I don't even know what the show is, lol, but I hear it brought up all the time when Ren/Wren is, and I know Ren was a guy...). I think Serena nn Wren is lovely, albeit a bit of a stretch. The negative reactions put me off of it, but maybe you won't get the same response. I think you could even do Wren Serena--the sound is different enough.
It's too much of a stretch for me, if she came up and said her name was "Wren" well I'd assume that's her name, full name, if she mentioned it's a nickname, my first guess would be Winifred.
Rena/Rina would be the nickname I'd pick for Serena.
I think it would be kind of hard to stick to it since it's not an obvious or common nickname for Serena.