Without venturing into the humorous side of this post, which is well-covered in other answers, I'd like to offer some alternatives, which could be less ambiguous.
Tim is a six-month-old photographer. (= He is six months old.)
Kate is a baby photo photographer. (= She is specialized in baby photo shoots.)
Pete is a nude model photographer. (= His profession is taking photographs of nude models.)
James is a nude female model photographer. (= Same as Pete, but more specific.)
Jake is a naked photographer. (= He doesn't wear any clothes while shooting.)
Jake is a photographer wearing no clothes in shooting. (= Similar, just more mouthful.)
The above examples only demonstrate some possibilities how we can be more specific while still phrasing our idea as a noun phrase. The last example (a photographer wearing no clothes in shooting) is still a noun phrase, but it's similar to Jay's example: The photographer, who takes pictures of females, received an award. Obviously, we can be more specific about our ideas in many ways. Usually, the longer, the more specific.
Language is about communication. In conversation, we talk cooperatively. (Thus writing is harder. According to Stephen King, "Writing is telepathy.") The cooperative principle describes how people interact with one another. Jeffries and McIntyre describe Grice's Maxims as "encapsulating the assumptions that we prototypically hold when we engage in conversation".
One maxim, Maxim of Quantity, says that we should contribute only what is required, not more. In general, your listeners or readers would understand your baby photographer and nude photographer just fine. Why? Because it's the most logical interpretation in most circumstances.
Grice's Maxims allow us to be concise. (In fact, one of the maxims, Maxim of Manner, requires us to do so: be perspicuous; be unobscured, be unambiguous, be brief, and be orderly.)
On the other hand, as noted in Wikipedia, "Many times in conversation, this flouting is manipulated by a speaker to produce a negative pragmatic effect, as with sarcasm or irony. [...] The Gricean Maxims are therefore often purposefully flouted by comedians and writers, who may hide the complete truth and manipulate their words for the effect of the story and the sake of the reader's experience."
Likewise, our listeners or readers can deliberately choose not to cooperate. As you note that baby photographer could be understood as a photographer who is a baby, rather than a photographer who takes photographs of babies. Tim is a six-month-old photographer can be reinterpreted as Tim only acts as if he was a photographer; he just plays with a camera, and never takes a single photo shot. Or in Kate is a baby photo photographer, it could be reinterpreted as Kate is a gynecology doctor who takes gynecologic ultrasonography regularly.
It's true that too few words can result in ambiguity.
It's also true that too many words can result in inefficiency and being less effective.
Thus, it's reasonable to try to strike a balance between using too many and too few words.