How difficult will it be to learn these concepts without the prescribed linear progression?
It's not really a linear progression, not at all. All of these subjects actually deal with the same thing. Mathematics is all connected.
Algebra (as it's defined by educators), geometry and trigonometry are taught separately because someone down the line thought it is the best way to teach. It's not. Those subjects just offer different methods/instruments to solve the same problems.
However, you don't really need to know everything that's taught in geometry/trigonometry by heart. Mostly it's all just some particular theorems which are hardly ever used in practice.
What you really need is to have a good grasp on the connections between all of these subjects. For example, all of these complicated trigonometric formulas are derived with the help of algebra together with a small set of rules, connecting $\sin$ and $\cos$.
Any geometric problem is eventually reduced to an algebraic one, and any algebraic problem (for example, solving an equation) can be represented in geometric terms. Not to mention, that $\sin$ and $\cos$ are usually first defined through their geometric meaning.
An example: the general solution for a cubic equation can be defined in radicals (with lots of square and cubic roots) or it can be defined in trigonometric form, which looks much neater.
Now, in practice, if you want to study a new subject and are not sure about the prerequisites, just start learning (through textbooks or some other way) and see if you encounter some unclear concepts or methods you don't know. Then use the internet or the help of a tutor to see what you need to learn/remember to understand them.
That's what I do when I need to solve an unfamiliar problem now that I am not longer a student: I just try and search for new methods if needed. It's a fine way to learn, but then again, I do have years of formal education behind me, which helps to recognize the connections I have mentioned.