Is the dydd at the end of gwledydd anything to do with day? Sorry if this is a stupid question!
No -- the -ydd is just a plural ending (one of many possible plural endings in Welsh, and the one that this noun happens to use) and it causes a change in vowel from gwlad (a country) to gwledydd (countries).
So there's no dydd "day" in there -- the d belongs to the root gwlad and the ydd is the plural ending; it's just a coincidence that they are together.
Gwlad, the singular of gwledydd comes from Proto-Celtic *wlatis (“sovereignty”); from Proto-Indo-European *h₂wélh₁tis ~ *h₂wl̥h₁téy-, from the root *h₂welh₁- ("to rule; strong, powerful"), of which also come Latin valeō ("I am strong", whence Romance valer/valoir/valere and English value). In the Germanic branch, some descendants of the PIE etymon are English wald ("to govern; power") and Walter (originally meaning "army ruler"), German Gewalt ("force, power, control, violence") and Swedish vålla ("to cause"). Other Indo-European cognates include Czech vlast (“homeland”), Polish włość (“small farm”), Russian власть (vlastʹ, “power, authority”) and Lithuanian valda (“land property”) and valdyti (“to rule”).