My birds wake up before I do, so I haven't experienced my alarm waking them. I guess I'm lucky that they don't scream in the morning, but they definitely shift around a bit (probably because the sunlight penetrates through the sheet I use as cage cover) before I greet them and give them fresh veg etc.
However, they do respond to the various tones on my phone/computer, usually with curious little squeaks! They're actually pretty chill; the only noise that really startled them was Slipknot, so I avoid playing super rage-y metal around them. All birds have different personalities and needs, but with some luck, it could be that if your alarm isn't blaring (if you set a phone alarm to a pleasant jingle) it won't scare them. Depending on your bird, you could also try covering their cage a little less, leaving a bottom portion or a side of the cage uncovered so that it's not pitch black in there. This way, they can see and are less likely to hurt themselves if woken.
TL/DR, have a "pleasant", non-blaring phone alarm. Maybe have a gap in the cage cover to allow a tiny bit of light in so your bird can see.