It sounds like you're a very dedicated parent, and it's clear you care about your child's learning. Also, your support of the instrument practice is amazing.
Juggling work, after-school activities, family life, and schoolwork is incredibly demanding. You've already got such good advice and support here, so I'd just like to add this:
Given your DC1 is only in Year 2, this will be mostly about building foundations for a habit of practice. So, 5 minutes of times tables and 5 minutes of spelling per day will be enough, but consistently every day. Short bursts of consistent practice are very beneficial at this age.
If you can make this a bit of a routine/habit, that would be ideal.
It's really helpful to give children about 30 minutes of rest after school or swimming/after school activities before starting on times tables and spelling. Ideally, playing or reading, not on a device. That gives the brain time to reset. This break actually could be used for the violin practice, as music does help the brain to relax and reset. Since he has shown he can learn with his instrument, that is a great strength to build on.
If you manage to make this a sort of routine/habit, it will be already in place when your DC2 joins in with homework in September.
You can turn the times tables practice into a game to make it more fun: "Let's see how many 5 times table questions we can answer before the timer runs out!" Use music and rhythm: As you mentioned, he loves music. Could you create or find a simple song or rhythm to learn the times tables? Maybe he can learn to play the song on his violin.
For spelling, there are some activities that children enjoy even if they are not on apps. For example, rainbow writing – write each word in a different colour of a rainbow, or pyramid spelling when you write the words starting with one letter, then add a second, then a third, etc. (e.g., the word school: s, sc, sch, scho, schoo, school).
I hope this helps.
It's understandable to feel worried by the school's expectations. But you are already doing a great job by trying to change things for your DC1 and these small changes will make a difference. Don't hesitate to reach out if you have any questions or just need a listening ear.
Good luck.