I wish people who were not rock-solid with their information would refrain from giving advice. OP, much of what was said above is incorrect.

With your UK passport, you get visa-free transit between international-to-international flights at Chinese airports. All of them where this sort of connection is possible. Connecting in Shanghai (PVG, SHA) you have 48 hours grace period for visa-free transit. Connecting in Beijing (PEK) or elsewhere, you have 24 hours. With this oddball double-Chinese-transit situation where there is a connecting domestic leg, you can still have visa-free transit at PEK. Your arrival/departure times are within limits.

You may leave the airport. Let's repeat this, heck let's shout it out: YOU CAN LEAVE THE AIRPORT AT PEK. I do not know why this misinformation is going about in spades. You cannot just "stay airside" at PEK like you might in many other airports in the world, you will be going through immigration although there are specially signed desks for "international transit" which is where you need to go to show passport and onward boarding pass/itinerary, and get the transit stamp in your passport. Once you pass this point, you can roam where you like, as long as you get back and depart within 24 hours. You'll be stamped out at exit immigration on your way back airside.

You may stay anywhere you wish, even off-airport. The lounges inside the airport I've had conflicting information on, even though I've been in there as late as midnight for a late CA flight--as to whether they are open 24 hours or not. Last I heard it closed 2-5 am. The airside "transit hotel" is expensive and often full up. The "time rate hotels" in each terminal on the nonsecure side are of varying quality but T1 and T3 are rumored to be somewhat better than T2. You buy blocks of hours according to your needs. You can stay at any terminal you want. T1 and T2 are connected by walkway. T3 is on the other side of the runway and requires airport free shuttle bus to go to T1/T2. Walk-up only, they don't take reservations. Nearby hotels off-premises are an option but the nicer newer ones may be poor value for just a few hours.

There is plenty of information on this on the China forum.