Ok, so I finally found out my own answer. It seems there are three options:
I went into the big Apple Store here in London, and asked a member of staff what I could do about a cracked iPad. The helpful staff member told me I needed to book an appointment with a technician. "Great, let's do it", I said. "No, you have to do it online", he replied. I was a bit incredulous. "Even though I'm here now, in the store?", I asked. "I'm afraid so", was his response. This seemed pretty silly to me, and to make matters worse, he said the earliest availability to see a technician was in a week's time.
So I duly went home, and booked my appointment online for a week's time.
Seven days later I trundled back into London, with my broken iPad in tow. I got to the store at my appointment time and asked a staff member where I needed to go to see a technician. They told me to talk to one of the Apple staff upstairs.
Up I went, and I found a bunch standing around chatting to each other. Eventually one of them decided to help me. The first thing he asked me was why I needed to see the technician in the first place, and so I told him the sorry story of my two month old iPad. One three feet tumble onto my bathroom tiled floor and it was cracked. Without looking at my iPad, or even mentioning a technician, he just told me that they could only fix it if I paid. And the cost for any fix, big or small, was £256.44. (That's $406 US dollars, American readers!)
I was a little taken aback and remarked that it seemed pricey. (I could buy an iPad Mini for that price!) "Yes", he agreed, "do you want to go ahead?". Apparently the only way I could talk to a technician was to agree to pay the fee now. That was all they could offer.
Needless to say I was pretty upset at having to go through all this rigmarole only to be told by a member of the floor staff that there was nothing they could do -- either I could pay, or I could live with it. They could have saved me a lot of time if they'd been upfront about it.
On the whole, it was a pretty terrible customer experience which left me feeling incredibly frustrated.
(Apparently Apple Care+ covers these sorts of accidental mishaps, but they don't offer it here in the UK.)
I've since found a few third-party repair companies that offer to repair broken iPad screens for various sums - between £70 and £110. So I'm considering them. (Some of them offer 12 month guarantees for the work they do, which is pretty good.)
I've also discovered that you can buy repair kits for between £30 and £40 that would allow you to do it yourself. From the videos I've seen, the job looks fiddly, but not impossible.
This appears to be all the options available to someone who's accidentally broken their iPad glass! :(