So many places depending on other factors. How old are you? Do you want luxury or don't mind roughing it a bit for example? Do you like interesting places or rather stay in western countries? Do you want to stay in one place or travel around (guessing as you don't like sitting on a beach it'll be the latter).
One option - at the very top end out your budget depending on what you do there would be somewhere like Namibia. Flights are about £800 and you can then do a group tour which will take you to the top of your budget. Slightly cheaper but in the same vein would be South Africa (cheaper flights). You could also do a mix of the two, fly into cape town and go into Namibia. Stunning scenery and superb wildlife viewing. For that price you'd probably be camping (which is an experience in itself, the campsites in Namibia aren't generally fenced so elephants and other animals will wander through camp) and you'd be going on an overland truck with long drives between locations (but again a great way to experience somewhere like Namibia). Not a goo d option if you'd rather be sitting drinking wine at the bar every night though.
The weather will also be good, the end of the dry season so little rain, but warm days and cool nights.
Alternatively somewhere cheaper would be Costa Rica. It'll be the end of the wet season and the crowds won't be there yet. You'll see more birds than animals (except monkeys).
There's also Canada as mentioned. The Rockies in September are great and temperatures are usually pretty good, but there could be snow, or it could be 25 degrees, or both on the same day.
Hire a car and travel round and depending on where you go there are lots of big mammals to see.
There's also somewhere like Yellowstone National Park. An absolute mecca if you want to see wildlife. You'll pretty much be guaranteed to see all the wildlife you'd see in the rockies (and more) but you'll have much more chance of seeing them.
All those are on the high end of your budget though!
Could look into Gambia, cheap and you could stay in a nice hotel for the duration and do day trips to see various things. September is wet however, when went there it rained off a canoe trip, but then the next day it was perfect for the day boat ride we did. Again, more monkeys and birds though, and you have to deal with all the hawkers as soon as you step out of your hotel. Also not sure if it's stabilised after the election there? Also not sure if it's worth two weeks.
Also there's places like the Camargue in france - flamingos, birds and wild horses. Not sure when the best time to go to see them is however, but you could hire a car and drive around southern france for a couple of weeks. There's tonnes to see and do there - Nice and Monaco a couple of hours east (and lots of little old hill villages that are must see's), the Ardeche a couple of hours North, Arles next to it (complete with various roman ruins), etc. That would probably be cheaper than the other options on the list. Flights to Nice can be had for £100.
