Does the time of Misheyakir in the morning get affected if it's overcast. Because then technically it's not Misheyakir.
According to at least one opinion, yes it is though I don't know exactly how seriously to take this source.
What is the earliest time you can make a bracha on your talis/tzitzis? Of course, the zman is Misheyakir, as in OC 18, when you can discern between undyed wool and tcheiles, or when you can recognize a casual acquaintance from a few feet away. When is that? The shittos vary from 66 minutes before sunrise (Pri Megadim, but very few hold like this le'halacha) to 60 minutes before sunrise (see Taharas Yom Tov, VII, 92) to 35 minutes before sunrise (Rabbi Moshe Feinstein, LeTorah ve-Hora'ah, no. 3, p. 7). Of course, you should consult YLOR. But if you follow Reb Moshe, your should know that Reb Moshe’s shittah is that just like a blind person is pattur from tzitzis, Misheyakir depends on the actual, local, subjective misheyakir, and if it’s cloudy, then it’s later.
(I added the bold).
The time of "Misheyakir" in the morning is NOT affected if the sky is overcast, much the same as "Shekiya", halachic sunset, is not affected on a cloudy day.
These are "technical" times, which relate to where the sun is currently located, in relation to the horizon.