While it's true that it might be a stretch to call Cusco's November weather “great,” this is perhaps the best month of the year to visit, and the weather is quite good anyway. The high season for hotels in Cusco ends in October, yet the rainy season doesn't begin until late December, so November is that perfect combination of small crowds, low prices, and very good weather.
Also, as far as I hear, the worst that ever happens is they take your stuff. So if you leave your passport and most of your money in your hotel room, and keep your phone or camera hidden most of the time, there isn’t much to lose. I’m making it sound worse than it is really. I wouldn’t hesitate to go to Rio again, but of course I’d be aware of those few situations to avoid.
From England I would say your cheapest and best late December destination with warm weather and reasonable prices is going to be Phuket. There are some other Thai islands that might be a bit cheaper, but they would be more complicated to reach, and they’d have fewer activities for the boys. Here are some Phuket resorts good for families. You can see that the price range is wide from cheapest to most expensive. In late December it’s peak season so you’ll pay at or near the top of those ranges.
Thank you so much for the kind words and I’m always happy to hear that people find this information useful. As far as Turkey in December is concerned, I wouldn’t recommend it. I actually lived in a small town near Antalya on the Mediterranean for a bit over a year and I was surprised by the fact that almost everything closed down around 1-November and didn’t start opening again until March or April. Actually, I think Istanbul would still be enjoyable in December and so would Bodrum or any of the other larger cities including Antalya. But anywhere on the coast will be a ghost town and I’m pretty sure that Cappadocia would also be mostly shut for the winter. Ephesus probably stays open, but similar to Cappadocia, the attractions are all outdoors and they can get pretty fierce rainstorms in winter.
One of the best things about Puerto Vallarta is that, unlike most of the Caribbean destinations, this is a real historic town rather than a purpose-built resort community. That means that there are small and authentic (and cheap) restaurants catering primarily to locals, so you get a much more authentic experience if you prefer one. The hotel zone close to the airport is modern and posh, but you are a short taxi ride away from the lovely center of town.
If you’re a bit more on the timid side, you can always use Yelp and TripAdvisor to get good insights on places people have visited. As a disclaimer, some of the ratings may not be accurate on these sites, so make sure you go through and read some of the more recent reviews that people have written. We did this for a few of the places we visited and had no issues.