Probably not much.
The colour of stars isn't that variable. A red giant is reddish white. A blue giant is bluish white. Even a red dwarf is a dark orange. Your eye has maximum sensitivity in the green.
In addition your eye adjusts to changes in illumination. That's why in a room lit by tungsten lighting you don't notice that the world looks like a photograph shot through a Holsum bread wrapper.
Tungsten light corresponds to about 2700 K which is cooler than a red dwarf Most people don't notice it, except that things that in daylight are a vibrant blue or green in tungsten lighting look a lot darker.
Take a look at a house plant. Your eye adjusts because your brain remembers what the houseplant is supposed to look like. You can have interesting discussions about fabric colours. If you haven't seen a clothes article in daylight, it's sometimes hard to say what colour it is lit by tungsten.
The blue of our present sky is mostly raleigh scattering. This is roughly linear with frequency, so higher frequncies get scattered more. The average is the blue we see.
A blue sky at 90 degrees from the sun has a colour temperature of 12,000. This corresponds to a stellar class B or A Yet if you look at something lit only by the northern sky, it looks more or less normal. (Artists like northern light because it's fairly constant over the day.)
A red star would mean less blue in the light so it would be shifted further away from the blue. However I suspect that it would mostly appear just to be darker.
If you get dust in the air you get a more pronounced difference. Dust being larger than air molecules scatter more, and also absorb more. See the copper coloured sun when you have forest fire upwind. But with dust the scattering is more uniform with frequency. The sky will be a smeared out version of the colour of the star.
You will also get interesting shadow effects. If both stars are in the sky, you will have two shadows. The shadow of the red star is illuminated by the blue one, and vice versa.
You can get a bit of the feel of this by going out on a sunny winter day. Compare where the sun hits the snow to a place where the snow is lit only by blue sky. The snow shadows are quite blue compared to the sunlit snow. This can be very pronounced in photographs.