Your son sounds harder work than most children, this is from a parent of at least 2 with ADHD.
My youngest is 4.9. High energy, stubborn as a mule, but if you take him for a run he gets tired out, he actually responds to things like time out and if I can catch him in a calmer mood after some misbehaviour happens and explain to him why I was cross, he looks genuinely guilt stricken and spontaneously apologises. Then we tend to see less of the same behaviour (he gets caught up in the frenzy of his older brother).
Neither of my elder two would do that - first of all, they wouldn't have a "calmer mood" after misbehaving. They were either in whirlwind mode, rabbiting on 19 to the dozen about whatever random thought had entered their brain or they were arguing about something or investigating something they shouldn't. The times that they would sit still were few and far between unless it involved a screen, or something they were incredibly deeply interested in, and even then not always.
You could not catch them to talk about their behaviour, because by the time you had finished the sentence their brain had moved on and they will have done 3 more forbidden things. "Six impossible things before breakfast" seems to be their motto in fact. They will have completely forgotten the first thing so there is no chance to get them to learn anything from the behaviour. It is really, really hard to keep up with a child like this. There are ways to do it without shouting but they all require a lot of front-loading. You have to be several steps ahead of them at all times. In the moment, something like shouting is often the only thing which has a hope in hell of working, but if you're relying on shouting constantly then essentially a klaxon ought to be going off because this is a warning sign that something isn't right.
Shouting did kind of work, and was the only thing that really worked a lot of the time. However it does also fuel what they call in psychology a coercive cycle. This is because the child knows they can escalate their behaviour and you will end up shouting or possibly even lashing out and then you feel guilty and stop asking them to do whatever it is - so they escape the demand. But on your side, sometimes it does work where nothing else does, so you feel like you have to shout because it's the only effective option. This actually reinforces the reaction on both sides. The second level of this is that because it's reinforcing the challenging behaviour you end up entering a state which they call blocked care. That means that you start to see your child's behaviour not as a normal aspect of childhood or even a sign they are struggling, but you see it as a personal attack on you and start to feel that they hate you or they are malicious in intent. It's extremely draining because it basically takes all the good parts away from being a parent as well as being a generally terrifying thought to have. It's highly unlikely to be true, but that doesn't stop it from being scary. You can get back from here, but you need the right support/advice. Keep reaching out and explaining how you are struggling. It's not your fault.
I am not saying your son has ADHD, because nobody could diagnose from a MN post, but if you relate more to the description of my two older DC than the younger one, then you might find advice aimed at parents of ADHD children to be helpful, because at the very least, your child's behaviour is extremely intense, and so 90% of the standard advice is going to be completely useless to deal with it. And NO, the advice is not "breathe a sigh of relief because you don't have to do anything now you have an excuse". Actually unfortunately, if you have a child with challenging behaviour you have to work twice as hard for half the results. But you'll be working 10x as hard for about 1% of the results if you are randomly choosing parenting approaches based on what works for other people, without tailoring it to the profile your child actually has. And TBH, there is no harm in following ADHD specific techniques for a child without ADHD. They are not indulgent or unusual techniques, they're mostly just standard parenting but applied in a much more consistent and proactive way than the vast majority of people have to stick to. The vast majority of parents can wing it most of the time with a little bit of pre-planning. If you have a child with very intense behaviour, you can't. And you'll be trying to not-wing-it on the dregs of the energy which are left over, which sometimes feels impossible, but it is worth it for how much conflict it reduces day to day.
You didn't say what the problems are at school but whether these are social/emotional, behavioural, or academic difficulties it is all a piece of the puzzle. Don't wait for the school to tell you what they think the problem is, as they are usually not supposed to be that direct, they are not doctors. But also because IME when you are at breaking point, you get a lot of side eye where people think it's parenting which is the issue, because the parenting looks very chaotic, whereas actually the fact you're out of resources with parenting is more likely to be a symptom not the cause. Be honest in how you are struggling because other techniques don't work, not because you haven't tried them.
Do you have a local Family Hub? I do not live in the UK any more, but my old local area has one of these and they are always advertising that they offer advice and drop ins and they are set up for general parenting advice but also topics like challenging behaviour or school difficulties. If not here, then GP, perhaps a different one (making an appointment about your child, not yourself) or HV, since they are still responsible for your younger child, or SENCo at school. Even if a parenting course seems pointless it can be a box to tick to say "I have tried this and it does not work".
Do you have a partner? You do not mention one, but you sound burnt out and you mention a lot of times that you are a better parent to one child, so divide and conquer if at all possible is a really excellent emergency technique. Screen time may also work for this, if it doesn't tend to exacerbate things (sometimes it does). Again emergency measures. If it is possible (and does not exacerbate) to extend time at school/nursery for either child, this also might be worth looking into.
Essentially if you want to break the patterns, it is going to result in some short term disruption for things like the hypothetical scenario where you spend so long dealing with DC1's behaviour that DC2 doesn't get dinner or to go to bed on time, unless you can have another adult available to take over. However I promise you this disruption genuinely is short term, you might be able to prepare for it (e.g. getting some instant/microwaveable food stocked up for short notice use) and if you can manage to get ahead of DC1's behaviour then staying ahead of it is not really more difficult than the current state of constantly running to try and catch up.
Honestly I still shout and I can't imagine a world where I would be so perfect that I never did. It seems to be a go to stress response for me. But it's much much less frequent, it doesn't even happen every week, let alone multiple times a day. DS1 I had no idea what I was doing and I feel very sad about what that did to our relationship, although things have improved. DS2 came along 10 years later and I have more tools etc to handle things with him. DS3 is fascinating just to show me how different all children are.
DS1 did get easier at age 6-7 ish despite me getting it all wrong, so there is hope. And if you can find the right thing then it might get easier early on. If nothing else, at least warmer days are coming and perhaps you can bundle them outside to a park? I found sometimes that was easier somehow. You will get there. It's OK to have times where you do not like parenting very much. I think everyone has their stressful period, whether it's newborns, toddlers, teens, or whatever this mess is (because I found the age 3-6 to be especially difficult with all my DC).