This has been irking me for the past month after I had to change my google password and couldn't get logged in on YT, and I finally had lucky logging in! In case you haven't had any success yet, I'll go through real quick what I did to get in, but I can't guarantee it'll work for you.
- I exited and restarted the YouTube app completely
-Went to login, and then IMMEDIATELY pressed sign in as soon as it popped up, SOMEHOW that miraculously brought up the keyboard! Make sure to only tap the blue "sign-in" button once, or it might make it keyboard pop up quickly and then go away!
- Typed my email into the box when the keyboard showed up, then pressed DONE, not GO. I think this was the one step I was getting wrong for days and days.
- After pressing done, press sign-in again. Not sure if I just got super lucky here when it went to the password screen. Before this, it would be stuck loading, but it's worth a try for you if I could get it to.
- Then type your password. I think you can just tap the info box to pop up the keyboard again, but if not, try pressing sign-in again.
And BOOM, miraculously it worked for me. Maybe sometime down the line someone can make a tweak fixing this login issue, but till then, this is the only thing that has managed to work for me. Hope you have the same luck!