BTW, the newel post, spindles and railings are being replaced.

Which edge profile really goes with your house best? If your house is colonial, then even if you don't love the bullnose, it's the best choice. And the match to your floors is great. The other thing to keep in mind is that these smaller projects can have a cascade effect. If your interior doors and baseboards are still colonial style, then once you make the stairs really modern, they're not going to look right and then you'll find yourself doing all that work too. It looks like your baseboards are modern, but I can't tell what your doors look like. What do the kitchen and bathrooms look like? How far are you going with your updates? I don't think everything needs to be done at the same time, but if the styling isn't consistent when you're done it's unattractive.

In your situation, I'd probably do neither and carpet the stairs instead. Not a runner; fully carpeted. It's definitely less expensive than matching the teak and will look better, IMO than mismatched flooring.

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The other option would be to paint the treads white and add a runner:

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There are practical reasons for a bullnose on hardwood stairs. For one, a square edge is much more prone to splintering and damage and repair can be more involved (vs a rounded edge than can be sanded & eased to blend). A rounded edge is more comfortable ascending vs a hard square edge cutting into your foot. Also, it looks more bulky and, IMO, the build-up on these particular oak treads looks much too bulky.

If you use a runner on the stairs, it is going to wear more quickly on a square edge than on a rounded one.

That said, the red oak doesn't look bad with the stairs. If you use them, then I would use red oak in the upstairs hallway.

The vinyl treads look surprising good, at least in the photos. If you use them, then use the matching vinyl in the upstairs hallway.

I would be inclined to use the vinyl treads (even as much as I love hardwood) because the match looks so good.

The LVP looks surprisingly good and that's probably what I would do. While I similarly like the edge of the oak treads, I think the matching LVP looks better. I don't hate the oak, so if your heart is with that choice, go with it.

I'm also surprised how well the LVP matches. I too would prefer bull nose over a square edge on stairs for various reasons, although that's not nearly as critical if you're also adding a runner. And that's just my personal opinion not necessarily rooted in any particular "fact" either.

The wood on my stairs doesn't quite match my wood floors. I too upgraded to wood stairs years after my floors were initially installed. They tried to match and it's close but not quite.

At first I was mildly annoyed but I don't even see it anymore. YMMV of course, but I know for me I tend to obsess over details during construction that get lost in the haze of life after they're done. That is a long way of saying either option you show looks fine to me in the end.

I would not want vinyl next to hardwood, ever.

I like the contrast between the darker floor and the lighter oak stairs - it looks intentional, and is far superior to "almost but not quite perfect match" (though I agree with the others that the vinyl color matches the floors surprisingly well). You can certainly get oak tread with a bullnose.

As you well know, dark floors show every spec of dust - not so much with the lighter color on the stairs.

When we replaced our carpeted stairs with hardwood in our previous house, we pulled out the treads and replaced them with full treads. I would rather do that than stair covers like you're showing.

We bought our house in 2019 and before we moved in, we replaced the original carpet on the stairs and also the entire upstairs with hardwood that matched the downstairs engineered wood. We also had the downstairs engineered wood sanded and stained. It looks lovely. I really didn’t think of anything else for the stairs other than matching hardwood. I believe the stairs were about a third of the total cost.
I love the continuity; however, I am always cautious when waking down the stairs. I often ensure one hand is free to catch the railing if needed. Sometimes my dog comes flying by. Sometimes I’m rushing down with socks then remember to slow down. I have slipped a few times, but always catch myself. I know someday that I’ll add a runner or something for better traction. That project helped me to now always make safety and function the priority.

Carpet is not an option for us. We don't want carpeted stairs and the hardwood or vinyl on the floors on both levels.

Also, AnnKH, why wouldn't you want vinyl next to hardwood, especially if it is on a different plane?

Either look good. I prefer the vinyl with the bull nose over the squared off other tread. I am not fond of the white risers, however. They will show marks all the time, from bumping with toes of shoes, etc etc. If you really want white, then make sure its a good quality paint that is washable, because, you will be washing them often.

I don't care for real wood next to imitation wood. As I said before, I'd replace what's there with solid wood treads.

Regardless of the material you use, make sure that the person doing the install checks that there is consistent rise at the top and bottom. When we replaced the upstairs flooring and stairs in our previous home with hardwood, the wood floor raised the living room floor by 3/4". The rookie they sent to put in the stairs did not account for this, so the top step was higher than the others - not only a code violation, but a serious safety hazard (which is why the code exists, of course). The owner got to come back and do the stairs over - at his expense.

Adding a layer - wood or LVP - to your existing stair treads could cause the same problem, though the higher step would be at the bottom. If that's the case, and the treads need to be changed anyway to solve the height discrepancy, you might as well put in solid treads.

Yes, you should check your state/ local code as to allowable variance in tread height. In my state (Wisconsin), up to 3/8" variance is allowed. So long as the vinyl overlay tread is 3/8" thick (or less), they would be allowed in Wisconsin.

ci-lantro, that would add 3/8" to the rise at the bottom step, and subtract 3/8" from the top riser - for a variance of 3/4". Also if the bottom tread is already at the maximum riser height, nothing could be added to it.

Be very careful with the rise height. The cretin (or maybe just cheapskate) who built our house carpeted the top half of our stairs and put oak treads on the lower half.

When we bought, we had the carpet removed and replaced with oak. That contractor wasn't much brighter, and we had two risers the wrong height: in the middle and at the top.

Had to get a finish carpenter in to re-do the top half so the risers would all be the same.

Ann, yes. But the flooring upstairs is being changed so that 3/8" can be dealt with.