For Sephardim:
According to the Kitzur Yalkut Yosef (Siman 46, Halacha 9): If someone is up the entire night, he must say the entire birchos hashachar except for al netilas yadayim and asher yatzar.
If he uses the bathroom, he does say asher yatzar but not al netilas yadayim.
This is different than the MB (who most Ashkenazim follow) in the following ways:
1. According to the MB, you would not say Elokei Neshama or hama’avir (as mentioned by the OP). According to the Yalkut Yosef, you do.
2. According to the MB, you would not say birchos haTorah unless you slept the day before. According to the Yalkut Yosef, you do, even if you didn’t sleep the day before.
3. According to the MB, you do say al netilas yadayim if you use the bathroom. According to the Yalkut Yosef, you don’t.