If it's a big enough hotel, they probably have guests leaving early in the morning. It's in their interest to get your processed and out of their hair before the big rush later on. The maids too, can't clean all the rooms at once so they usually "attack" those rooms which are free as soon as they can.
Another option, bringing up the good shower suggestion above, is once they allowed us into another room which wasn't yet cleaned, handed us some towels, allowed us to dress and get out. Thank and tip them profusely and perhaps keep this in mind when packing, leaving a complete outfit for each near the top of the bag or in a side pocket to whip out if necessary.
The hotel can then hold your bags and away you go... People... lights... action...
Just the buzz for them being in a new place should give you all a bit of momentum. Remember to buy snacks to put in the room. Your stomach might want dinner at 4am. Wont happen but a rumbling tummy will keep sleep at bay. Something diet-unfriendly, like bland cookies or a bagel (in London??) Have some munchies and get back to bed!
No T.V., computer or video games before bed or if anyone gets up in the middle of the night. It supresses melatonin, what we're doing now. Read up on good old melatonin if you're curious, or just take my many years of crossing that ocean's word for it!