When I got back into photography after a break from film, I went with a Nikon d90 + 18-55 + 55-200 and found changing lenses in the field a real hassle. If you can go for a Nikon with the 18-300 f/3.5-6.3 DX VR, it would be a really nice kit. There's also an older 18-300 f/3.5-5.6 (the older is faster at 5.6 but heavier) that is very good too. There's a Tamron 18-400, but I've not heard good things about it, so I'd stay away.
Having said all that, the Nikon 70-300mm f/4.5-6.3 DX VR AF-P lens is very light, inexpensive, and is supposed to have a fast auto-focus. I've not used it. If you don't mind swapping lenses, or will be using this most all the time, it's probably a great choice for wildlife. Because it is a DX (not full-frame) lens, it's a good weight match to the d5000s.
Canon does not have anything like this and their 70-300 lenses are full frame so will be a fair bit heavier (losing any advantage of the tiny SL2/200d). I think many Canon shooters go with the 18-135 STM plus a long lens, as the 18-135 is a very good walk-around lens that won't leave you short like the 18-55.
As for the body, the d5500 is about the same thing and price as the d5300 but a bit lighter and adds touch screen. The d5600 (current model) is about the same weight as the d5500 and adds bluetooth/NFC, which isn't much. The d5300-d5600 all have the same autofocusing mechanism. So, if a few oz of weight matters try the d5500 instead, otherwise they are all the same photos. The SL2 is a new Canon (2018 I think) and also a very light body (a fraction lighter than the d5600). It takes great photos too.
Personally, when I'm out shooting birds I bring a Tamron 150-600g2 ($1300) plus my iPhone. No short zoom. You can probably find the last generation of the 150-600s (Tamron or Sigma) in the $650-800 range and the current set in maybe $1000 used. The tammy is really worth the $1300, IMO (I've not tried the Sigma). If you want to shoot birds, I'd try to find a way to get to at least 400mm native if not 600mm native.
To sum it up, if the main point is to shoot birds, I'd try to get the Tammy or Sigma 150-600 and either Nikon or Canon, as you please. A canon SL2 (200d) + 18-135 + 150-600 would be great, for example (or the d5x00 + 18-55 + 150-600). If you cannot swing the 150-600, I think the Nikon d5x00 + 18-300 or 18-55 + 70-300 AF-P is good too. What I find helps me is to make a spreadsheet with all the weights and prices on it, so I can balance cost and weight and reach.
Hope that all helps! I think there will be little quality difference in the photos between canon and nikon bodies, it is all about the lens combos and their weights and minimizing the need to swap lenses.
Marc