They are normally 4mm or 5mm, but probably 4mm. Your nut looks very deeply burried, but there definitely is a nut in there. You may just need a longer key. Try using the long handle end of a hex key just to see if it fits in. You may need to get a hex key with two long ends. You can find them on eBay or Amazon, often labelled for acoustic guitars that have a body-end truss rod adjustment that has the nut inset a long way from the soundhole.