At 3 o’clock in the morning I was awoken by the crying of the baby. I had to pick her up and give her to her mom.
Um 3 Uhr morgens wurde ich vom Schreien des Babys geweckt. Ich musste sie vom Bett aufheben und sie ihrer Mutter geben.
I am not sure if I can also say “ich wurde beim Schreien des Babys geweckt” in this context.
No, “Ich wurde beim Schreien des Babys geweckt” would deny that the crying baby is the cause of being woken up (like “I woke up during the time the baby was crying”).
Also remember that Germans prefer compound words:
Um drei Uhr früh wurde ich von Babygeschrei geweckt.
I am surprised to see so many posts confirming the expression vom Bett aufheben. To me it sounds rather un-idiomatic, and I'd prefer aus dem Bett nehmen or simply hochnehmen. (Note that the English sentence doesn't mention a bed at all.)
Additionally, in German you could also say um 3 Uhr in der Nacht or um 3 Uhr nachts. I am not sure whether at 3 o'clock at night would be possible (and idiomatic) in English.
And last but not least: IMHO we do not use das Baby very frequently to speak of our own children. (I might be biased in this point, but at least I want to mention it). It is more kind of a neutral term, a word that can be used to speak about very young children in general.
To sum it up, and to make a point for not always sticking too close to the original, I would propose
Um 3 Uhr in der Nacht bin ich aufgewacht, weil die Kleine schrie. Ich musste sie aus dem Bett nehmen und ihrer Mutter geben.
Passive by translates as von or durch. Beim would translate as while.
Also, Baby is neutrum, so the second sentence should contain es rather than sie:
Ich musste es vom Bett aufheben und es seiner Mutter geben.
You would rather say “Ich wurde vom Schreien des Babys geweckt” instead of beim.
What you want to express with this sentence is, that you was awoken by the crying of the baby. Therefore you use the word vom/von dem, which is the correct translation of by.
Um 3 Uhr morgens wurde ich vom Schreien des Babys geweckt. Ich musste sie vom Bett aufheben und sie ihrer Mutter geben.
This is the way you say it in German.
Your suggestion is wrong. Beim schreien does not translate to by (the crying of) the baby but rather to by crying. So your suggestion would mean:
I was awoken at 3 a.m. by crying of the baby …
That is just a great mess of weirdness. Immediately, the suggestion to correct it pops to my mind:
Ich wurde um 3 (Uhr) in der Früh beim Schreien nach dem Baby geweckt …
This ‘correction’ sounds closest to what I would expect with beim schreien; however, it means by crying at the baby. So we have the paradox that it was you crying at the baby that woke you up … you see how this is just getting more terrible by the minute?
I assume that your inital assumption was something along the lines of ‘bei is a preposition, too, isn’t it? Can’t I add it to that noun there?’ Yes, we can possible think of grammatical ways to use bei with a noun in the context of waking up, but in that case it would be the place where you woke up.
Ich wurde um 3 in der Früh beim Brunnen (vom Nachtwächter) geweckt …
Note that the original construction is passive voice using a by-agent in English to denote the active agent. In German, the active agent is denoted by von in the passive voice.
Furthermore, I wish to restate, that picking up a baby would be much better translated as hochheben or hochnehmen rather than aufheben. Aufheben is something you do to things that fell, usually on the ground. We don’t want our baby falling, do we?
