Does the same apply to typing on a computer?

In my opinion, no. Typing on a computer is not writing, nor is it enduring in the usual sense (rather, it exists in memory). It seems to me more similar to speech than to writing. It is certainly not professional craftsmanship.
People commonly assume that this is writing that counts as ordinary, non-professional work rather than skilled work, but as I said, I doubt whether it has the status of writing at all. See here: http://ph.yhb.org.il/12-11-13/

According to this, would it be forbidden to type on WhatsApp during Pesukei DeZimra (or at another time when an interruption of speech is forbidden)?
Maybe one could say that it is like thought. Another practical implication would be whether one is obligated to recite the blessing over Torah study before writing Torah words on a computer.

I think it is speech and not thought. The thought is being actualized here through an action. And indeed, it would be an interruption just like speech. Perhaps one could distinguish and say that only ordinary speech constitutes an interruption between one spoken passage and another, but simply speaking, no.
As for the blessing over Torah study, it seems obvious to me. Granted, according to the Vilna Gaon in section 47, one recites a blessing even over thought, but here, in my humble opinion, it has the status of speech, as above.

If it is like speech, then maybe one could type Pesukei DeZimra on WhatsApp and thereby fulfill one's obligation…

I imagine you understand why that would not work…

And if writing is like speech, then in any case it would also work with ordinary writing. See, for example, here:
http://olamot.net/shiur/%D7%9B%D7%AA%D7%99%D7%91%D7%94-%D7%9B%D7%93%D7%99%D7%91%D7%95%D7%A8
See there that even if writing is like speech, that does not necessarily mean it is identical in every respect.