I have photographed a lot of birds with an Olympus E-M1 II and a Panasonic 100-300 II. It should be easy to find the body and lens for under £1000 on the used market, and the pair is just under 1100g.
If you can spend more, the OM-1 is a significant upgrade because it has bird detect autofocus and a bigger buffer, but if you're not always at a long distance, extra money might be better spent on a second lens. The Olympus 12-40/2.8 is outstanding.
