As CWS says I'd not leave until the last minute. I fly on a ski trip via LGW every year and check my Skis in the night before, but I think the latest I've done this is 18.30 or 19.30 in the oversize luggage. One time there was no one there but BA page them and the chap came over. So don't panic if no-one is around. All pretty seamless.
If you are staying at LGW I've used the Bloc hotel on a couple of occasions and it is okay. I've used the Hilton exec rooms and they are poor value. Indeed the way to them is the back of the hilton through the stinky corridors of the normal rooms. I was bored one day doing and overnight and I timed coming from the Hilton room and compared to coming form the Sofitel in the North terminal and the Sofitel via the wee train is faster. Premier Inn at LGW North is also very good.
I know you only asked about bag drop but I assume you are staying overnight and thought the extra hotel info would be useful