Learn English – Do Canadians and Americans really use “gotten” as past participle when speaking

The Cambridge History of the English Language, Vol 6, confirming what Barrie England wrote in his answer, says:

In tracing the history of American grammar, it is important to note that in many instances what might seem to have been a change in American English compared with the standard English of Great Britain is in fact no change at all - it is British English that has changed, not American. For example, eighteenth-century speakers of English generally formed the past participle of get 'receive' as gotten, as in Your brother has gotten my mail. In the nineteenth century, prestigious speakers in England began to drop the -en ending: Your brother has got my mail. Most Americans, however, continued using the older form gotten.

I had it taken recently is definitely NOT in the past perfect!

I took it recently = You did it yourself.
I had it taken recently = You asked someone to do it for you.

(Both sentences are in the simple past.)

BACKGROUND

The name "past participle" is a rather unfortunate one. It's one of the sources of confusion in English grammar learning. This is mainly because the English passive voice and the English perfect use the same form of verb, i.e., the past participle form of a verb.

I quoted an entry in a grammar book once, in this answer (follow the link for more information). To demonstrate this point with the same examples, I'll discuss them, briefly, below:

Past participles are used to make both passive verb forms and active perfect tenses. Compare: (1) He was calling. (active - past progressive)
(2) He was called. (passive - simple past)
(3) He has called. (active - present perfect)

In (2), called, the past participle form of call, is used after the auxiliary verb BE in its past form, was. A main verb in its past participle form after BE indicates that the clause is in the passive voice. In this example, someone called him -- he was called -- he was called by someone.

In (3), called, again the past participle form of call, is used after the auxiliary verb HAVE in its singular-present form, has. A main verb in its past participle form after HAVE indicates that the clause is in a perfect tense, in this case the present prefect. In this example, let's say that someone ask you, "Jack told me he'd call the lieutenant today. Has he called?" and because Jack has done so, you may reply, "He has called."

BACK TO OUR QUESTION (BARE PASSIVES, e.g., have something done)

The confusion caused by your sentence, I had it taken recently, is perfectly understandable, because in this sentence, had is the main verb, NOT an auxiliary verb. This is a special use of have, as defined in the Macmillan Dictionary in definition 14, like this:

have
[transitive] [never passive] to arrange for something to be done or for someone to do something
have something done: The place is looking much better since they had it redecorated.
She wanted to have her portrait painted by a famous artist.
have someone do something: I’ll have someone bring your luggage up right away.

So, I had it taken already means that you asked someone to take the photo, and the photo had already taken.

One way to explain the syntax of have something done is to treat it as a bare passive that complements the main verb (typical verbs of that allows bare passive complements are have, get, order, and other sense verbs such as see).