Yay for another Egyptian!! I have a Straight Egyptian Arabian gelding. To give you an idea of his size and growth - at 2 years old (on his birthday), he was 13.1hh. As of 2 weeks ago (he'll be 3 at the end of May), he was 14.2 1/2hh. He's still very petite (his breeding).
I started mine lungeing right around 2. I slowly progressed the length and difficulty of the lesson. I started with normal lungeing. Once he was really good with that, I moved on to adding a surcingle. Then I added the bridle to the surcingle. Next, we worked on bitting up. Then I started with the saddle without the bridle. That's where we are right now. We currently switch between saddle and bitting up. Now, all of this was over almost a year and I never progressed unless he was solid at the previous step. In the past month I've finally felt that he's at the right size and build to start letting my trainer/best friend lay across his back - and it's only happened twice. The first time, she just laid across his back on both sides. The second time, she used more force to lay across his back on both sides. He'll be 3 in 1 month and 10 days and I know he won't be built enough to be ridden. Weight-wise, he's 800#, so technically he can support both of us, but he has these super petite legs! He makes a great halter horse, though.
With Egyptians, they're so fine-boned and petite that you have to wait simply because they have to fill out a lot more before they can be ridden. It's not so much that they have to mature because as far as muscle and bone structure go, they mature right along with any other Arabian (and most other breeds for that matter). Just keep building the muscle. On the lunge line, I do a lot of fast trotting to help build muscle. Of course, my guy is bursting to the seams with energy, so he also does a lot of cantering (he's always had crazy endurance), too.
I didn't mean to ramble. His growth depends a lot on what he's fed and the size of his parents. I don't expect mine to reach 15hh (I'd be delighted if he did, though) because the parents were 14.3hh and 15hh. So, if he's small and petite like my guy, bet on waiting until close to 4 years old more than likely. Around 3.5, you'll be able to walk him around and MAYBE lightly trot him (depends on what kind of a mover he is), but just be aware of how he's filling out and growing. That's just through my own experience with my Egyptian and with other Arabians and Egyptians that I've worked with in my life.
The wind of heaven is that which blows between a horse's ears. ~Arabian Proverb