This happened to us tonight. I'd had this apartment booked for months and they had organized a car to pick us up from the airport to drop us at the hotel.
We arrived in Rome, driver picked us up, dropped us off and assured us someone would be there to check us in. 5 minutes turned to 15, and I was starting to worry. At 30 min I was making calls, but no one from the apartment owner was answering.
Called booking.com however their phone system didn't make it easy. Took ages to get through to a human, and They couldn't contact the owner company either.
At this point we just found the nearest hotel, (three doors down) they had one room for three people, so checked Emma, Sue and Sharon in, as Emma was exhausted and needed sleep.
Paul and I went to work organising something suitable for the week. Booking.com came through with a replacement apartment for the same price, not that far away. So Paul and I are here tonight and will move the rest of us in tomorrow.
Remaining positive about it is the key I think, the new apartment looks fine and is in a great location.
Will post more updates about wtf went wrong with the first booking when I know more.