VelvetSpoon I totally understand where you are coming from regarding the school. Very similar here. I am guessing that your DCs dont have a discipline problem in the classroom so a bit of lateness probably seems very minor to the school.
I also understand about the no formal bedtimes. We dont and somehow my DCs have survived!
For your oldest how do you teach self-discipline? I'm not sure you can teach it though you can facilitate its learning.
First off, how much are you talking to your older DS about the future? I dont mean 'big chats' but an ongoing dialogue. Does he see the next step as a reality, something which he has to strive for? At the next stage he will be the one who has to put in the effort. The consequences of lateness will fall on him directly.
I think the mid-teen years are actually quite hard for a lot of boys. They are being expected to take responsibility in terms of getting to school, doing homework, behaving themselves. Yet, they often just dont seem quite up to the job yet. The 6th form stage is very difficult if they havent got their act together. I think it comes as a shock to many to discover that they have to sort themselves out, that no one comes chasing them if they are late or dont do work.
For your older son there has to be an acceptance by him and you that he is very close to being a grown-up. As a grown-up if you dont get up on time the consequences are generally slower to arrive but when they do they are far more draconian. I am not sure how much I would be punishing him. He isnt a little boy anymore. He is now at the age where being sorry or taking a punishment wont make it all better.
Talk to him about developing good work habits. Talk to him about long-term consequences. This is one good thing about a poor school, lots of very real life examples of what happens to the people who dont sort their lives out!
For your younger son, he is old enough for something like Army Cadets. This has been great for my DS (now 15) as they do place an awful lot of emphasis on self-reliance. Also, my experience is that they do just 'get' teenagers, especially boys. When my DS joined he wouldnt put his hand up for anything. Now he is going for promotion.
Your younger son can still benefit from 'Mum' discipline. Early bed times with lights out. Electronic devices off early. Are you monitoring homework? In year 8 I found it was very much like being a sports coach. Continuous low-level nagging coupled with riotous enthusiasm for minor achievements!
I know that this is all very hard. I work long hours so find that a lot of my parenting is done at weekends.