How do these decimal number displays work? I guess there are assembleys of a mask with the digit, a lens and a lamp for each of the 10 digits. So each of the very beautiful number fonts are projected to the backlight screen. It is amazing how sharp and precise these images are.
I'm not sure about this precise type of display (*), but there are ones very similar and that exactly match your description: Ten individual light bulbs, ten lenses and a mask with cut-outs for each digit.
They were called "in-line displays" or "projection displays" and became popular around 1960, before they vanished in favor of LED-based 7-segment displays.
Here you can find a video of one of these displays "in action": https://imgur.com/gallery/59GNG8m
Here's also an dismounted device with all components visible: http://www.industrialalchemy.org/articleview.php?item=511 (As there is no license given, I'm not going to repost these images here)
(*) They look slightly off to me, on the image it seems they are just reflecting light and are not lit by themselves. Could be that this is a different type, or the original displays have been replaced for the exhibition. Here's an original image of the control room, taken from Wikimedia Commons:
In the bottom left corner you can spot several numeric displays like these, but they are definitely lit and as to be expected from these kind of display, the digits are not as well aligned as the ones in the rebuilt.
I'm pretty sure that the numbers in RETRO's chronographs are stickers (or nice LEDs), but not connected to actual clocks etc (the leap-second gives it away).
The original photo from Wikimedia Commons shows much more clearly what look to me to be Nixie tubes. The slight misalignment of the baseline between numerals when seen off-axis is a strong indicator, as each digit would be at a different depth in the tube.
Given that NASA procurement was on government contract to lowest bidder (mostly), I doubt NASA would have used something as expensive as in-line displays. In addition, the mentioned in-line displays would be less reliable (predictable life) and require more maintenance than standard Nixie tubes that had been used for decades already.
My vote is Nixie tubes in the original, and stickers in the restoration display.