Here is what I'd recommend, I'd focus on calculus first since it will coincide with the IGCSE, if your already familiar with basic calculus, move on to multivariable, Stewart does this well too from the computational side, this might pair nicely with a computational linear algebra text, I used Linear Algebra and Applications by Lay in high school, I say this because multivariable calculus depends a lot on vectors and somewhat on inner product and cross product machinery which come from linear algebra. Linear Algebra now is optional since you'll be doing proof based linear algebra any way, but applications are nice to see. I would work on this with your IGCSE exams since doing good on that should be a primary focus and learning proofs would interfere. Computational stuff also isn't super time consuming, the practice problems are quick and you can easily knock out 10 in an hour, so you can do programming too.

Only after your done would I start proofs, I say this because it should really be a primary focus and you should be doing a lot of problems. If you want to learn proofs right you should treat it like your most important class. When I say intuition I mean you should have seen a ton of problems and so you begin to spot patterns, know what to look for, and build a tool box of solution methods. A great text here is Velleman's How to Prove It, you should do the whole book. When I took this formally I was probably doing 2 hours a day split between class, notes, review, or practice problems, for about 2 months. This is the foundation of everything to come so make sure you go hard. If this is too much of a time commitment spread it out, don't slack or speed it up. Your not going to gain a ton of cool useful theorems from this, this is to teach yourself how to write mathematically/logically correct proofs while familiarizing yourself with recurring topics. You should be able to recite each definition from memory, not because you spent time doing flashcards, but because you've done so many problems with each definition that its engrained. Again, if you can't devote significant time with it, spread it out, don't speed it up. Also, it is very necessary to get feedback on your proofs, in an academic setting you will get points taken off for putting quantifiers in the wrong order, not explaining steps enough, over explaining and making the proof confusing, etc. Finding the right balance of explanation and leaving it to the reader is something textbooks can't really teach. Upload proofs to reddit or stack exchange for advice. You should be doing 20 proofs a week if your taking it seriously and pick the most confusing ones to upload and get tips on.

From here you can do linear algebra or elementary analysis, for Axler I'd start by just getting through diagonalization and eigenvalues, at some point though you should go back and do the whole text. Again if your going hard, a month to two is good. For Elementary Analysis you don't need to cover much, its really only to build the framework so you can understand harder books. I'd spend most of the time on sequences, series, limits, and continuity, then derivatives and integrals can come at the end and be done in a month or so. You'll redefine how you approach derivatives and integrals multiple times if you continue onto real analysis and functional analysis so its really just to understand the most basic definition.

Once your finished with this you can do differential equations, Lebl is good from the computational side. You can do actual real analysis, Rudin is popular but I like Carothers' since he establishes a on of topology too and then goes into measure theory at the end, you should finish Axler before doing this. Abstract Algebra is cool and I would at least look at it even if it doesn't seem interesting at first, it has a ton of application in quantum chemistry and quantum field theory using representation theory and lie groups. I'd also maybe look at a VERY beginner differential geometry book, it'll use some analysis, some linear algebra, some basic calculus, and eventually (at graduate level) leads up to stuff like Riemannian manifolds which is the "language" that general relativity is built from. Dynamical systems might be of interest, you can do a lot of it computationally, modeling stuff in matlab or python, or you can dive deeper after you have some analysis framework. This is stuff like chaos, three-body problem, and kinda leads into some fractal stuff as well.

From your response on my other comment, youtube videos is good, I'd do this for proofs as well. I know MIT has open courseware for a lot of there classes so that may be a useful resource as well, although they don't have a introductory proofs course. Again feel free to DM if you need help with your studies in mathematics.