Will Avogadro's number change if definition of atomic mass unit is changed?
Simple and actual answer:
The current definition of Avogadro's number doesn't depend on the unit of mass. They are completely independent quantities. The Avogadro's number is currently simply defined as $\pu{6.02214076e23}$. It's just a number which is important enough to get a special name.
The answer you are probably looking for:
Historically the Avogadro's number was defined in such a manner that it did depend on the unit of mass.
At that time the Avogadro's number was defined as the number of atoms present in $\pu{12 g}$ of $\ce{^{12}C}$ isotope of carbon.
Now as you can see even if we defined an atomic mass unit as $1/24$ of the mass of an $\ce{^{12}C}$ atom instead of $1/12$, it won't make any difference to the Avogadro's number as it was defined on the basis of grams of carbon and not atomic mass units of carbon.
Now coming to your second question of whether this would have changed the mass of a mole of a substance. Yes it would have changed that, all the molar masses would have been double of as known currently.
Note:
All this discussion is just about how someone wrote a definition and the loopholes someone can detect from it using language.
For example if we really would have defined an atomic mass unit in this different way, we would have also defined Avogadro's number as number of atoms in $\pu{24 g}$ of $\ce{^{12}C}$. Even $\ce{^{12}C}$ would have been written as $\ce{^{24}C}$!
The takeaway is that the scientific idea and origin of all these units was(and is) always there  but it's just the nuance of defining it in such a way that it doesn't create such problems was the problem. That's the only reason all the standard units were restandardized recently. It's a sad reality that schools are still teaching the old definitions even after we have said them a good bye.