I agree with Tor that your 4970 will bottleneck a 2080. And overclocking wouldn't solve it. However, with that said you would still see a good performance gain over the gtx970.
As I said earlier I just upgraded to a 1070ti and a ryzen 5 1600. I would have liked to go with a Ryzen 7 2700x but did not have the extra coinage to make the full leap. That will happen in a few months. My 1600 was even bottlenecking the 1070ti, until I overclocked it to 3800mhz. Now they are happy together lol. To give you some numbers, with the 970 and FX8350 I was at 64.2 FPS average and a score of 2688 on Unigine Valley. Min max was 18.3/120.3. After the upgrade those numbers were 74.4 average with a score of 3112, min/max of 22.9/156.5.
Performance increased but not as much as I had hoped.
However, when I OC'd the Ryzen to 3800mhz (on water-cooling), my numbers jumped to 94.8 average, score of 3968 and min/max of 31/169.9.
A ton faster but my processor is still bottlenecking my vid card. That will be solved when I get a Ryzen 7 2700x.
So Binak, if you have the money to buy a 2080 it would definitely improve your system performance considerably over a GTX970, but you would want to plan for a future upgrade on the processor, and your max path with what you have would be the Ryzen 7 2700X i'm eye-balling.
I'm thinking you would be better off to use the money to get a new motherboard, low end threadripper and like a 1080ti. That would set you back a few hundred more than just a 2080 but you would be way ahead of the upgrade curve. But if you currently have DDR3 (I'm guessing) you would also have to buy new RAM (DDR4). So I suppose what you need to think about is if you want a video card your system can't take full advantage of until you upgrade the MB and processor/RAM to a Socket sTR4 motherboard and something like
https://www.newegg.com/Product/Product.aspx?Item=N82E16819113457. Combine that with a 1080ti.
A 2080 will cost about ~$700-$750. New MB, a threadripper, 16GB Gskill DDR4 ram would be about the same, then toss in another $650 ish for a better vid card. I don't know what import duties you have to pay though. I bought my 1070TI used off of EBAY from a miner for $300. He had 4 of them never OC'd but with about 6 months 24/7 usage. I've had it now for a couple months and it still works great.
The jist is you will have to upgrade both eventually to get the full performance so I suppose you just need to determine which upgrade to do first.