Hi
Well you've obvviously done a lot of homework and I always applaud those willing to travel to Jaffna and the north, partly because it's so fascinating and different from the rest of the island, partly because the people there are often among the poorest and really need to get something back from the growth in tourism (95% of all accommodations are in about a quarter of the island, in the southern hill country and beaches - says it all!). The north is also the only part of the island where road transport is relatively fast because (a) it's generally flatter (b) the roads have been rebuilt after almost total destruction in the war so they're much better.
OK, on to some specifics. First, the weather. Yes, September is generally a good month, especially in the east and north, but not always. I've kept very basic weather records over the last four or five years and September here in the hills is the most unpredictable month of all - can't speak for the coast or dry zone though.
Arrival - if you're travelling non-stop from UK you'll be on UL504 which arrives around lunchtime. If so, you could go straight to Galle same day - with the expressway it's only a couple of hours' drive from the airport. Colombo itself is not that interesting - except for great eateries and shops. Historical and other attractions are limited in my view. But if you're on a later flight then yes, it makes sense to have a night in Colombo and head on to Galle next day.
If you have only a night in Galle that's enough to explore the historic Galle Foirt area but not any of the beaches or other places around. But if that's your main interest, it's fine.
You shouldn't need two nights in Uda Walawe. It's a great park for elephants and the Transit Centre is very much worth visiting and supporting but you can arrive in time to visit that, do an early morning safari next day and head on. Especially if you visit Minneriya/Kaudulla later, which I'll come to.
You may or may not want to spend three nights in Ella. It still gets billed as a traditional Sri Lankan hill village but the reality is it is crowded and one of the most over-touristed places in the country due to the endless hype. The scenery is tremendous but the walks you mention are those that everyone does and you won't get the hills to yourselves (if that matters). Also Nuwara Eliya is not really a day trip from Ella, it's a long journey. So I would suggest one or two nights in either Ella if you're set on it or Haputale if you want an equally lovely place with few tourists (but also less choice of accommodation), then train to Nuwara Eliya for a night, then on to Kandy either by train or hired vehicle. And spend two nights near Kandty but not in the heavily polluted city itself. From there you can do not only the main Kandy sights (Temple of the Tooth, Peradeniya Gardens) but also explore the Hanthana and/or Knuckles mountains for more off the beaten track walking, and learn about the history of tea (Ceylon Trea Museum, various working tea factories, historic Loolecondera estate) as well as hidden gems like the Three Temples near Peradeniya (much more interesting, cheaper and less crowded than the over-hyped Temple of the Tooth IMO).
Then I would skip Kalkudah and head first to Habarana (via Dambulla and Sigiriya). From Habarana do a safari in Minneriya because September is the time for "the gathering" in Minneriya - the biggest concentration of wild elephants on the planet. You might also want to see Anuradhapura or Polonnaruwa from there, and the lesser known but staggeringly beautiful Ritigala. So you may need two or three nights.
Then on to Trinco where the beaches north of town (Uppuveli/Nilaveli) are every bit as good as those in Kalkudah, plus the attractions of Trinco itself. Have your main beach time there. Then on to Jaffna where again you'll need a minimum of two nights as there is the town itself but also the weird and wonderful offshore islands like Delft to explore.
If you have time between then and your flight home (haven't done the maths!) you could stop for a night either on Mannar island (the best place for birding in this very bird-rich island, if that grabs you) or visit Wikpattu national park if you want more wildlife - few elephants but leopards and sloth bears as well as crocs, four types of deer, lots of birds and stunning scenery. Then south to Negombo and the airport.
How does that sound? Do send a personal message if you want more details and ideas.
Have a great trip
Jerry