How critical is an uncoated lens? Color photography was developed before lens coating. Take a look at old, well-preserved color photos, and see the effect of an uncoated lens. Do you actually see a problem?
Contrast lost through lens flare is always a problem. Somebody posted a link to a test where a box with a test target in it was photographed against the sun. The test shots were made with and without a lens shade. The photographs with the lens shade showed much better shadow contrast and detail.
What you need to do is make tests for yourself, and see what works for you.
Unfortunately film development can't compensate totally for the lack of contrast due to internal flare with older uncoated or poorly coated lenses, I include some Multicoated lenses in that statement.
Michael R is right about many 1950's coated lenses being almost equal in terms of contrast to modern Multi-coated equivalents, there's also a misconception that lenses have a single coating, some do but others have more than one coating layer Zeiss used 2 quite early on and some pre- Multicoated lenses were not far different in terms of their coatings.
Early coated lenses were problematic with colour films often giving quite a strong blue cast and warm up filters were introduced to compensate. Lens manufacturers overcame this in the 1960's with newer coatings and often these lenses were designated to reflect this, Color Skopar, Pancolor etc.
My experience using an uncoated 1930's 135mm f4.5 CZJ Tessar, a mid 1950's 150mm T coated 150mm f4.5 CZJ Tessar, a late (2000/1) coated 150mm f5.6 Xenar and also various Multi-coated 135mm & 150mm Symmars & Sironars is that apart from the uncoated lens the results are almost indistinguishable in terms of contrast even shooting into the light. In any light there's a drop in micro-contrast with the uncoated Tessar and that's the same for my other uncoated lenses, there's a loss of detail in subtle highlights. The 50's Tessar is very heavily coated and visually give a blue cast but it's remarkable good for B&W work and micro contrast which is imprtant for fine detail is as good as a modern MC lens, and it's very flare resistant.
The problem with discussions like this is not all manufacturers coated their lenses properly, they wouldn't coat every air glass surface or coatings were poor.
A classic for poor Multicoating was the Hoya range of lenses in the late 1970's or early 80's launched with a lot of hype. They were very sharp lenses but Hoya skimped on the coatings and many suffered from severe flare, the result was Hoya dropped the entire range and went back to the drawing board, the new range was launched under their Tokina brand name.
So the choice of lens does have an effect on the contrast and particulary micro contrast, highlight and shadow separation, varying exposure and film development time can only help to get the bst from the inherent contrast of the lens itself.
Regarding different comments about visible improvement from coating, I also noticed that single coated lenses were good enough at least for the type of pictures I take.
Lens contrast is very important. You cannot compensate for missing detail in the darkroom, or by increasing development. Single coated lenses can be excellent, multicoating is better but shows it's virtues more in complex lenses such as zooms.
One thing to keep in mind is the value of an efficient lens hood. I use lenses of all ages, from early 1900s to modern multicoated wonders. The hood helps them all. An uncoated Dagor with a compendium hood has better contast than a single coated Symmar without the hood. A multicoated Dagor with compendium has better contrast and more shadow detail than any lens I've ever seen, used, or heard of. My late 60s/early70s Nikkors 35mm lenses have all the contrast I or anyone needs, when used properly.
Veiling flare can be printed through in B&W, but can cause color casts in the shadow areas on color film.
Dirty lenses will show markedly less contrast than clean ones.
So, keep the lens clean, use a hood, and don't obsess over contrast. Any modern coated lens that is clean and in good condition - and modern means post WWII - will give adequate contrast for any purpose if used properly.
Any kind of light bouncing around the lens before reaching the film degrades image quality, some detail must be lost. You might like the effect, or not, you might even need it, but the lost "quality" (in the sense of information) is not something that can be recreated in the darkroom.
High internal flare reduces both the contrast of the lens and the "micro-contrast" (acutance) which is one of the qualities that we perceive as sharpness.
That is only of interest to those persons, or circumstances, when you value sharpness of course.
So my answer is that it makes a difference, and that it is a difference that cannot really be compensated in the darkroom. But that said, using a low-contrast (high-flare) lens and raising contrast in the darkroom can give you the results you were looking for.