あなたのTwitter教えてって英語でなんて言うの？

Asking either of these questions will be enough to ask another person what their username/ handle is on Twitter. Often times I hear people say, "You can find me on twitter @__".
I think both of these examples will be fine.

When you're trying to find out what a person's Twitter account name is, then you can say:
-What is your Twitter username?
-What is your Twitter handle?

The word "handle" when referring to social media usernames is usually used in casual conversation.

"What's your twitter name?" This asks the person to tell you what the name of their twitter account is.
Eg: I would like to follow you on twitter, What's your twitter name?

"What's your twitter account name?" This is another way of asking the person to tell you what the name of their account is.
Eg: I would like to follow you on twitter. What's your twitter account name?

You may ask, "How can I...?" if you want to know the procedure for doing something.
"How can I get to London before 4pm today?"
More directly, you may ask: "What is/are........?"
"What is the quickest way of getting to London before 4pm today?"

A twitter handle is the name given to your twitter username. This will be prefixed with an @ symbol.

Twitter is a very large website that has over a billion accounts and it continues growing! Because of this, it can be very difficult to find our friends, family, or other people we know on Twitter.

If we would like to ask someone what their Twitter account is, we can use some useful expressions such as:

1) "What's your Twitter username?"
2) "How can I find you on Twitter?"
3) "What's your Twitter handle?"