Learn German: Der Täter will das Geld nicht überwiesen haben. - The perpetrator claims not to have transferred the money.

This is the subjective use of wollen. When used this way, usually with a past infinitive, the subject is claiming they did (or didn't do) something. It strongly implies that they are making this claim about themselves, but the speaker or the police aren't necessarily convinced it is true.

How does the grammar behind überwiesen haben work?

This is called a perfect infinitive. Modal verbs like will push the main verb to the end of the sentence in its infinitive form. Because the perpetrator's claim is about a past event, we can't use the normal present infinitive (überweisen). Instead, we combine the past participle (überwiesen) with the infinitive of its helping verb (haben), which literally translates to "to have transferred."

Why is nicht placed right before überwiesen haben?

In German, nicht usually goes right before the specific part of the sentence it is negating. Here, the perpetrator isn't denying that the money exists (das Geld); they are specifically denying the action of having transferred it. Therefore, nicht sits immediately in front of the verb phrase at the end of the sentence.

Could I use überwiesen if the perpetrator handed the money over in cash?

No, überweisen specifically refers to transferring money electronically or sending it from one bank account to another. If the perpetrator physically handed over a briefcase of cash, you would need a different verb, like übergeben (to hand over).