My logic is that if Everton are at home on Saturday the BHA match has to be on Sunday for police reasons.
If it's televised (which I don't think it will be) then it'll be at 4 pm. If it's not then they'll go for the usual post-lunchtime 1.30 pm kick-off so fans can not drink too much before...
I don't think the broadcast contract includes provisions for broadcasting of the 5th round by ESPN or ITV on a Friday night, and unlikely the clubs would agree to this when Sunday is available.
That won't matter - ESPN and ITV will make their choices completely independently of what Sky have broadcast previously. And, they'll go for the ties likely to be most attractive to non-aligned supporters.
One of the Liverpool or Everton games will have to be on Sunday - those clubs are "paired" for policing and Merseyside Police will not allow them both to play at home on the same day.