I strongly suspect the bishop "hats" were swapped by a prankster. Bet you can separate them and swap them back?
How to repair this chess set
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If you want to shellac it, you'll need to do more than just clean it a bit. Shellac is not difficult unless you are in a hurry (I tend to apply very thin coats), its more forgiving and more repairable than modern finishes are. The question is how it will go over what is on there now (including the moisturiser) and thats potentially getting complicated and may end up looking odd unless you are much more aggressive with the board. A simple wax finish will make what is there now look and feel a bit better. I don't know either of those waxes, but look at the ingredients or the MSDS and pick something simple. The most basic furniture wax is just beeswax dissolved in turpentine with a few other things added these days to improve shine (which is where the carnauba comes in). Adding silicone or polythene is good for the strength and shine of the surface but starts down a track that does not necessarily lead to an easily maintained, easily removed finish.
Oh sure, it could be moisture, it could be sun bleaching, I have no idea. It looks like the veneer is still there. I just worry about trying to fix it. If the colour comes from different timbers that are just sun-bleached or the finish has gone white (water in shellac for example) then that defines one approach, if its sun-bleached stain thats a different approach. I'd worry that re-colouring the squares could lead to mismatched or too obviously repaired bits. I'd be very surprised if the colour was applied to each square individually, why bother with veneer if you are going to do that? its usually different wood to get the effect. To get down to the raw wood or remove a cloudy finish will take a more aggressive approach than cleaning it with a cloth and the rest of the board is looking more or less ok.
Wax may be minimal protection, but this is not a yacht, the board needs something that will stop the dust and dirt from sticking too much. If a chess board is getting wet, or bumped, or abraded (the use cases for high protection finishes) then its not in a typical environment for something old and of sentimental value. I agree, without a whole lot more work, shellac or anything else could be a bad idea.
@bo_mb 
No that is how it is supposed to be. This is Checz design. Although I am from Hungary, many old chessboards look like this. Check it out:
https://www.chess.com/forum/view/general/czech-club-play-chess-set-design-how-do-you-like-it
Here is a close up of the affected squares. Shellac / varnish is missing, veneer is still there. I can not really tell if these are two different kinds of wood or just painted differently. I think these squares a sun-bleached. Notice the difference in color between A1 and A3.
its an interesting question why A1 is so much darker than A3 (and a different shade to B2), and A3 looks almost the same as A4. The finish could be shellac, but if you were going to re-coat it you'd need to take the surface down further and with something in that fragile state (the lifting bits of veneer around the edges for example) you might damage it more. Could A1 be a replacement piece? Is it the same thickness as the others?
Interesting indeed. The thickness is the same. I am suspecting that it is not replaced, but was different from the beginning. Notice how the corner of the square is also missing. I think the makers were out of standard black squares, and used a left over one, possibly from a previous batch with a slightly different color. Look closely at the black pawns on #7. You will see little differences. They are not identical, but there are 3 slightly different kinds! Also there is a stamp on the bottom of the board that suggests that it was sold on a discounted price. I think we just found out why 
@FlyingSandal Mark, I am not yet sure about your idea with the dark wax. How would I do that? Apply a layer of colorless wax on the whole surface, and then the dark wax on the dark squares? Or apply colorless wax on the light squares and dark wax on the dark squares one by one?
I think it could spread the wax unless you were super careful. At this point you either need to decide to live with what you have or take a larger risk. Frankly, I'd live with a working but imperfect board. Waxing will improve the finish and provide a bit of protection. Shellac could work but you'd have to get the surface where the finish has gone clean and smooth because its got an annoying tendency to make imperfections more visible when you apply it over scratches you didn't notice, and then we are back to the problem of the thin and somewhat fragile veneer.
greghunt wrote:
Everything depends on your objective: stabilised and somewhat cleaner would be straight-forward but not without risk. I wouldn't paint the pieces.
Start by stabilising the box, gluing the board to the side of the box. Introduce the glue into the gap with something very thin, it doesn't need a lot but you need to try to cover the whole of the surface to be glued. Make sure you can wipe any excess glue off (PVA glue for example wipes off with a damp cloth, resin glues do not clean up so easily, CA glues set very quickly and don't wipe off well, hide glue would be good but is not extremely common), and that you have the right clamp to hold the board down to the side while the glue sets (make sure you have some pad between the clamp and the wood so you don't leave a mark on it. The clamp will squeeze some more glue out.
Cracks in the boards can only really be addressed by more major surgery, try to avoid that.
Its hard to see how thick the veneer is, sanding is a very bad idea with most 20th century veneers, they are too thin. A very fine (000 or 0000) steel wool with gum turpentine could be used to scrub the surface of the box clean and then be wiped off. Raw linseed oil works nicely and leaves a nice finish but I'd worry about the effect on the colour (turpentine will darken it but will dry out again). The risk is that the oil will darken the board and soak in unevenly (this would not be a concern with the sides of the box). You might use water, but that could affect the wood and the finish more than oil or turpentine would, but in different ways (the veneer glue might be water soluble, the timber might move with the added water). A tree-based turpentine may be the safest thing to start with, but it will not leave the nice finish that oil would.
The loss of colour is difficult to deal with without knowing how it happened. If the squares are different woods then cleaning may well fix the problem (damage to the finish could be hiding the colour). If the squares are dyed differently and the dye has been affected by damp, or they have been painted with different colours, or something, then thats harder without ending up with a really obviously repaired or replaced bit.
A finish for it: a traditional hard cabinetmaker's wax or possibly a pure carnauba car wax, both of which are removable with turpentine, (you may decide you want more a radical fix) applied first with fine steel wool to rub it into the surface and rub it off, and then a cotton cloth to remove the last of it and polish. This should be left set up for a few days. Carnauba is shiny, beeswax is more of a satin finish. Its not a really robust finish, but this is a chess board.
And finally the result after the waxing
Left is the original, right is the current state. I have tried to recreate the lightning conditions as close as possible. The colors are definitely brighter, the pieces are shinier. The big crack under the surface is fixed.
Thank you guys for all your help!
I'm glad it worked out well. If you re-apply the wax periodically the finish will get smoother and more even. It should feel nicer after the waxing too. I spent some time yesterday waxing go stones (181, one at a time) and bowls with a carnauba wax, its a kind of therapeutic thing to do.
I think this is a Romanian set
I don't know how much you want to repair/alter the board but you can buy complete chessboard veneer
https://www.ebay.com/itm/353455765430
If that board is a strange size, one can also try to diy the checkerboard veneer
(it's not as complicated as one might think since you glue long stripes next to each other, not squares)
https://www.hansiherrmann.de/tipps/schachbrett-bau.php
Mark, rest assured this will never look new.
However, I have checked some blogs about shellac, and they indeed seem to agree that it is quite challenging to apply. I definitely want apply some kind of finish on the board, because honestly right now it does not look like it can bear an other 60 years. So I think I will go by waxing.
Most of the web shops here sell Annie Sloan or Lignocolor branded waxes. Do you guys know these brands? Lignocolor is carnauba, that Greg suggested earlier. Is that good?