Help with size of bolts, please

Ooops, finger trouble in the cold. 6.35mm

While I agree, after the typo has been corrected, it does seem odd for what, from dates in the instructions, appears to be of relatively modern manufacture and where all measurements are given in millimetres. If it is an American import, I suspect they could be 1/4" UNF.

I was surprised when I bought a towel rail from China via eBay to find that the Allen grub screws were not metric. I suppose if a manufacturer can find a whole load of screws cheap, then it doesn't matter what size standard they comply or complied with.

Can you measure the thread pitch - as in how many threads per inch or whatever? That will help to identify what the tread is. The diameter alone is not sufficient. There are lots of cases where there are 2 or more bolts with the diameter but different pitches.

Does the OP have a camera (video, film or still) of any kind? The mounting hole will be 1/4-20 UNF. That is also remarkably close to 1/4" Whitworth, I believe there's only a small difference in thread angle between them, and they are interchangeable.

Not quite right. 1/4" Whitworth is very similar to 1/4" UNC - (Not UNF) Both Whitworth (BSW) and UNC are coarse threads, and the 1/4" variety both have 20 threads per inch (TPI)

The fine thread equivalents - BSF and UNF are less similar, with different pitches. 1/4" BSF is 26 TPI and 1/4" UNF is 28 TPI.

So the OP has quite a lot to choose from - hence my suggestion about measuring the pitch.

The thread could be 1/4 20TPI.

The Everbilt dowel nut has a Philips head instead of a Slot head like this one.

Paul