Arriving London early on January 1--where to wait?

You may get lucky with the weather and have a decent-ish day. If it's really too miserable to be outdoors, check the possibilities in this article, which indicates that there are at least a few museums open that day. (I haven't gone to the museum websites to verify that.) You'll need to find a place to ditch the luggage, though. Maybe Google will turn up a commercial luggage-storage option.

There are apparently also some stores open.

London Walks has a few walks on New Year's Day. They're only 10 GBP each, and you just show up, no pre-booking required. So you can make a weather-based decision at the last minute.

I would suggest trying Citystasher. It's a listing service where you can find businesses that store luggage for a fee. It is completely insured and not super expensive if you just need it for part of a day. New Year's Day may be a little difficult but many of the businesses that store luggage are tourist centered so you may find something available on that day. That way you can at least walk around and see things without dragging your things around with you!
VisitLondon has a good list of things to do on New Year's Day to give you some ideas. Costa Coffee opens at 6:30 AM normally but I think they open at 8 on NYD so at least you can have some breakfast and warm drinks before venturing out into the city.

The train stations are open on the 1st day of the year - at least if some aren't closed for HS2 or some other construction debacle - and several of the London main line stations have a left luggage counter. Not cheap, and they charge for every piece (you mentioned packing for 10 days like it was a lot to take - I hope you remember that your flat will likely have facilities for washing and maybe drying (hope so) so you all can pack really light). Humping bags around for nearly a whole day will get old pretty fast...

Have you asked the apartment owner or manager if you can drop off the luggage? We have never been refused that courtesy in many apartment rentals over the years and frequently the apartment is ready much earlier than estimated/listed in the confirmation.

I will suggest a museum for you to enjoy during your wait, The Museum of London.
It covers, obviously, the history of London, from pre-historic times to the Roman occupation to present time.
Very interesting, and free.

I have checked their website, and it looks like they are open January 1.
This may be a good place to visit while you wait for your apartment to become available.
The museum has lockers where you can stow your back pack and luggage. Lockers have a lock on them.
You insert a token, close the locker, take out the key to take with you. £1 per use. Purchase tokens at reception.
Maximum size 43x37x24cm. (17" X 15" X 9") Larger luggage cannot fit into the lockers.

There are two cafes in the museum and one restaurant.
This will allow you to take breaks which browsing in the museum and get coffee, tea, and a bite to eat.
A very pleasant place to spend the afternoon.

One note: My husband and I travel light, as Nigel suggested above.
If your luggage is too large for the lockers, one of your jet-lagged party who doesn't want to walk around the museum might wish to sit in the cafe drinking coffee, and guard the stack of luggage you pile in the corner.
I know I could easily pass a couple of hours that way, reading a magazine or the local newspapers.
People could take shifts watching the luggage until time to go.

Luggage that doesn't exceed 17" is traveling very light indeed.

Hi acraven! 17" is not actually that small. You can cram a lot of stuff into those bags!
The duffel that my husband carries to London for a 10 day trip is 17" long in the base.
The L.L.Bean tote that I carry for a London trip is 17" long across the base, 15" tall, and 8" deep.

We did the Rick Steves England In 14 Days tour this past spring, and these were the only two pieces of luggage we took. (Except I did take my purse.) The tour communications we received before our trip specifically asked us to each bring only a carry on bag. (No big huge pieces of luggage allowed on RS tours.)
We did fit about 15 pieces of clothing each into these bags. And we counted on doing a small load of laundry every couple of days.
Another alternative is to divide your clothes between a bag of the size we use, and a backpack (or perhaps a day bag) for each person.
I'm only discussing this because acraven (sort of) brought it up. People should take whatever size luggage they want to; it's a personal choice.

Are you sure your apartment won't let you stash your bags there? I'd email them and ask.

Thank you all for your helpful ideas! We will follow up on most, maybe all of them. I'm hoping the apartment management will allow us to stow our bags somewhere, but if not you've given us some possible alternatives. I can't wait for a cup of Costa coffee, and a visit to Museum of London.