LED emergency lighting - relay activated
1) You don't need the switch because if it's an emergency light Murphy will have turned it off at the most inopportune moment.

2) You don't need the delay because it's just adding a lot of unnecessary stuff to something that's supposed to just work and not have reliability problems.

3) You don't really need the 5 volt supply since you can just use a relay with an AC coil. However, it's probably a good idea to use low voltage where you can and minimize the possibility of someone getting zapped.

4) Make sure that, if it's a 5 volt relay, that 6.2 volt supply won't fry it. 

Having said that, here's how I'd do it: