Given the extended history that is complicated by time-travel, I think that any cultural shifts are possible, and so Gallifreyan could easily more closely resemble a conlang than a natural one from earth, not just in variety of phonemes, but in the more basic ways that language works. Tenses could be attributed to nouns, or given in two different forms. (One for the object's history, one for the observer, or more.) Letters could be attributed to several phonemes at once. (At 3 phonemes per letter, you can get over 10 million letters with 253 phonemes, and it could easily be more phonemes per letter than 3.)
I suspect that written Old High Gallifreyan had aspects involving the direction and form of the writing, such that one sentence could be written right to left, and another could be written like a mathematical matrix, and yet another in a form more like circular Gallifreyan; later the language would evolve into a formal and informal forms: One informal like the second picture, and one formal like the last two pictures. Often important information like warnings on fire extinguishers and console information is given in the circular form, and more handwritten or generally informal forms are simplified into what could even be right to left forms. I suspect Old High Gallifreyan could've had some letters that would not be pronounced, but instead only read to allow the reader to interpret the format of the text, similar to a riddle, but easy for Gallifreyans, of which we know are much more intelligent.
It's surprisingly possible that only one Gallifreyan language ever developed, even branching into different forms. Before the Age of Rassilon or the rise of the Neo-Technologists, communication was done entirely telepathically, making language irrelevant, so the Gallifreyan language would have to have been made, either during or just before the Age of Rassilon, for the entire empire. It could quite easily have made Latin look like Esperanto ten times over, being made be technologists of a species around twenty times (the Doctor's IQ reportedly is 2000, from one episode) as intelligent as humans.
That would actually mean Old High Gallifreyan (Should that start with proto-?) is a ConLang. To give you an idea of the difference between conlangs and natural languages, first think about all of the things that make English a very weird language that makes no sense, and then browse the ConLang subreddit for a while. They're more logical and modern, but can be much more complicated.