We are yet to receive the first light from most of the stars in the universe. Every second we receive light from new stars and this has been the case since our planet was first formed. Conversely, a toy example explains that if the universe was infinitely big and infinitely old, the night sky would be homogeneously white to our eyes.
Now. Given that the universe is not infinitely big but old enough for us to receive light from all the stars (with redshifts and all that), then how bright would the sky be?
We first need a reference. The color of the sky is not black. Black at the end of the day is not a color but a symptom of the lack of light hitting your eyes. So, how dim is the night sky? Thankfully, Pantone can answer that question directly. I trust the world's "color experts" to have a good ballpark approximation. For me, this seems good enough an approximation to my daily experience but for more detail, you can go here.
The universe started 13.8 billion years ago. A bit more because the record of it that we have is from some minutes after the clock started when the universe cooled down enough to be transparent to the cosmic background radiation. Yet, as it started, the universe expanded fast. A speck of dust located at the edge of the universe which moved as it expanded will be now around 46.5 G(light-years) away. I will take this number as the baseline for our calculations.
Let's make the objectively false assumption that stars are homogeneously distributed in the whole universe. Second, let's follow the Bible and assume that the Earth is at the center of the universe (sorry Galileo!). Naturally, we also assume that Heliocentrism has been the rule of the world since the Big Bang, and thus the universe is spherical with the Earth in its center. This simplifies our problem dramatically.
To find how much more light we would see we just need to divide the volume of the universe by the volume of the observable one. But since both are spheres we just need to divide the radius of the universe by the diameter of the observable universe and elevate this ratio to the third power to get the multiplier.: 306.
Now things get a bit more complicated. Turns out that not all the light that we see in the sky is star-light from outside the solar system. In fact, this light accounts only for 7% of the light we observe (somehow) the rest comes from Airglow and Zodiacal Light. Therefore, if we were to see the light from all stars our sky would not be 300+ times brighter after all. It would be around 22 times brighter than it is today.
To put this in comparison, the sky with the full moon is around 7 times brighter than with a new moon. So, if we could see the light from all stars in the universe, the sky would look as if the Earth had three moons shining on it at all times and way dimmer than during daylight Interestingly, the full-moon would still make a relevant contribution, yet just one-third of the nightlight would come from it.
Calculating the equivalent Pantone is complicated. Because, if we are honest, the "Night Sky" color is quite bright (13% to be exact). So, if we triple the brightness we would get something like this:
I am quite sure this color is wrong and way too bright, but it is still nice to imagine that if the sky was full with the light of all the stars in the universe, the most we would see is a normal grey sky. Yet, the sky would have some 300 times more stars.
Update 210705: I found another link that explains that "On a starry night, the illuminance is 0.001 lux. On a full-moon night, it reaches a maximum of 0.3 lux. The skyglow of a city... can reach illuminances of up to 0.1 lux"
From this data, having 300x stars would put the brightness of the night sky on par with the brightness of the sky when the full moon is present. In contrast to the full-moon situation though, the light would cast no shadows as it would be homogeneously distributed!
The results, although different, differ quite little when considering the back-of-the-envelope manner of my calculations. It seems that indeed if we see the light of every star in the universe we would see a brighter sky but we won't need our sunglasses to go out at night.