"Let me talk to ..."
"Can you get ... to the phone?"
"I want to talk to ..."

Hallo Alan,

danke für die schnelle Antwort!

"Let me talk to ..." ... ja, manchmal ist es so einfach, dass man eigentlich auch selber drauf kommen könnte ...

Melle

i find "let me talk to..." and "I want to talk to " a bit abrupt I have to say. I wouldn't say "can you get..."

Can I talk to ...
Is ... there?
Can you put ... on?

Naturally, you say something like Hello first along with your name when you make a call. "Let me talk to..." and "I want to talk to..." shouldn't be that abrupt then. However, they do carry some of that command tone of ""Gib mir mal ...".
Of course, the perfect snappy retort to "Can I talk to ..." is "How should I know about your ability to talk? Am I a speach therapist or what?!!"

Hi, Bobby, can you pass me on to your mum?

Perhaps it's a BE thing. I would never say Let me talk to... unless I really intended it to sound abrupt. For me, this is another of those occasions where the imperative is OK in German but not in English.

I would say:
Could/Can I speak to your mum?
Could/Can you put your mum on?
or the suggestions in #3 + 5