On Tisha b'Av and Yom Kippur, we are not supposed to lave our hands. Resultantly, on Tisha b'Av and Yom Kippur, do we say "baruch… n'tilat yadayim" in birchot hashachar?
This statement is inaccurate:
As we learn in the Kitzur Shulchan Aruch in סימן קכד - הלכות תשעה באב laving hands and getting wet in general is not a problem, if not done for pleasure:
Only washing for pleasure is forbidden, therefor one washes Negel Vasser in the morning.
Since one has washed, one obviously says the Bracha of עַל נְטִילַת יָדַיִם.
However, since "washing hands" - according to most opinions - is only until the knuckles, on Tisha B'Av and Yom Kippour one may not indulge in the luxury of washing until the wrist, (as one usually tries to do, if one has enough water).
[Since only washing for pleasure is forbidden,] if one's hands are dirty, one may wash the dirty parts. Similarly, after going to the bathroom, one may wash one's hands "a little" in the way one normally does. [IOW, if you always soap your hands after the bathroom, you may do so, but keep it to a minimum, concentrating on hygiene and no more.]
The Kitzur adds that even for Mincha one may wash one's hands [minimally]; to reinforce that it's not only for morning prayers.
Not only is hygiene and ritual washing allowed, but incidental washing is also allowed. Thus one is allowed to wash food [for kids to eat, or for after the fast], even though one's hands get very wet, since one is not doing so for pleasure.
If one travels for a Mitzva [to visit the sick or one's parents, or to go to shul, for example] and the only way is through a body of water, one may cross the water and get wet [if there's not boat available], both on one's way there, and even on the way back home.
However, if one travels through water for financial gain / business, one may go but not return through water.
Even if you use the third definition of lave
After a long trip by foot, if one's feet are sore, one may wash them with water.
All the above comes for Hilchot Tisha B'Av, but they are essentially the same for Yom Kippour, as the Kitzur says in סימן קלג - הלכות יום הכפורים :
On Yom Kippour only washing for pleasure is forbidden, The Kitzur then refers you to Hilchot Tisha B'Av for details.