Panny, you can't bag-drop unless you are checked in, so since one can OLCI 24hours before the flight departs, I would assume "any time before the flight" would mean any time after you have checked in (OLCI or otherwise). You must be in possession of a valid boarding pass, so OLCI will either give you a mobile BP or allow you to print one.

I'd rather keep my bags until the morning I'm flying, since then I have the "liquids" I need for the day of the flight. Hilton T4 to LHR T5 isn't too bad, and if you need help with the luggage, it should be no problem. I also think (though this is just my rumour) that the shuttle bus picks up on arrivals level and drops off at departures level at T5.