What part of speech are the words in the phrase “as well as”?

In the example you have provided, as well as functions as a coordinating conjunction replacing and. See:

• They stole my car as well as my laptop.

• They stole my car and my laptop.

The entire phrase is a synonym for and.

As well as is a substitute for and, so the verb were is correct. If as well as was, actually, a non-essential phrase (a parenthetical element that doesn't alter the meaning of the independent clause), and set off in commas, then the singular conjugation was would be the correct verb. As it is, the synonym for and (coordinating conjunction) takes the plural conjugation.

"As well as" is a prepositional phrase that functions like a conjunction. That doesn't make it a conjunction, technically. The result of the prepositional phrase means the same thing as the conjunction functionally, but it's literally saying that x is "as well as" y.

In that light it's not a conjunction, even though it has the effect of "and" if you choose to read it that way.

On the other hand you could understand it as a colloquial phrase that has been stripped of its prepositional value. In some cases, this may be the easier route... though not necessarily not so much correct. But, what's correct when the speaker doesn't know the difference?

As well as" is a prepositional phrase that functions like a conjunction. That doesn't make it a conjunction, technically. The result of the prepositional phrase means the same thing as the conjunction functionally, but it's literally saying that x is "as well as".

Precisely.