Android - Nexus 5 Reboot loop
I think you encountered the issue with a stuck power button. I had the exact same behaviour that also went away without explanation to return a few weeks later. I read the one trick is to bang it in a hard surface and shock the button out of its stuck state.
The second time it occurred I actually found a much more gentle way to temporarily fix this. Take an electric tooth brush without brush and push the oscillating shaft onto the backside and side of the phone close to the power button. After a few seconds of that my phone was back in working order.
And I felt strangely like Mc Gywer.
*Update: This temporary fix still works two months later. So it was not so temporary considering the lifecycle of a modern smartphone.
*Update 2: This fix worked fore a few years. Now the Nexus is permanently stuck in a reboot loop and no toothbrushes or other vibrating tools will fix it.
On an earlier version of Lollipop, after an update, my Nexus 5 would turn off randomly and enter a boot loop. I tried entering recovery just as you did but it too kept rebooting. I researched reasons for a boot loop and discovered some users of the Nexus 5 have faulty power buttons.
Try repeatedly tapping the power button as if to get it unstuck. See if the boot loop stops because of this action. If it stops, carefully hold your N5 so that the power button does not get stuck and turn on your phone. Perhaps jiggle the phone a little after the long press to ensure the button jiggles too.
It sounds simple but this reliably booted my phone when it had troubles. I installed Gravity Screen (free in Google Play) and try my best not to use the power button. I have not had many troubles since and am on a long streak for no issues.
I mention it happening after an update but I would guess the issue is just coincidental with the update. Best of luck.