We have four children (7 years between oldest and youngest) and no help: I waited until my youngest was in junior school (yr 4 - age 8) before I went back to work. I was only really able to do it because my husband had just been forced to go part time (20hrs) in his job so he picked up the slack I left behind in regard to household chores/cooking and children.
We managed because our children were all in local schools so the older ones helped out with school pick ups. Music lessons happened in school, after school activities weren't a big thing for us then, all the kids could swim, and older ones would help with getting our younger two to and from any local classes - but we found after-school activities tended to naturally die away once they hit secondary school, or it was things they could get themselves to. In fact all our children became much more independent at getting themselves organised and getting around our local area. We've called on local friends once or twice, but otherwise our lives just took on a different rhythm and everyone adapted.
We probably couldn't have both worked full-time to be honest, not without arranging childcare and a cleaner, but it's definitely do-able if one of you is working part time. To answer your specific concerns, here's how it worked for us.
We have no family support around: Many people don't, work with what you have rather than fretting about what you don't have.
Somebody would need to collect the younger DC at 3.15
Who would get the kids to school? Wait until the younger ones are not babies any more and use the older ones to help when you or your partner can't take/ collect.
I would need to take days off for sickness (with 4 DC they can tend to get things one after the next): I never really found this once they got past infant school age. There really was not that much sickness, maybe the odd day here or there.
What about school holidays? Luckily my partner works in a school, but I'm sure you and your husband could work something out.
They would need to drop after school clubs probably as it might be too much for a childminder to facilitate: Yes, but as they get older you'll probably find they'll be glad to do this.
When would I do food shopping and dinner would be a rush Plan ahead, use online shopping, work part time and food shop in smaller amounts as you go rather than a once weekly 'big shop'
I'm not sure I could keep on top of the house, even with a cleaner Get a grip
When would I fit all the other "stuff" in because I find I'm running round most days as it is Take a step back, if you disappeared tomorrow their world would still keep turning and the essentials would all still get done. If you're 'running' most days with four school age children and no paid job to do then you're doing jobs that don't need to be done, recognise that some that work will go by the wayside.
Would the childminder be able to facilitate / check homework because with 4 DC this is a lot? You'll be there in the evenings to help, homework gets done, or it doesn't and they get into trouble - that's up to them. I never found that it was a lot or that they even wanted much help from me after year 4 or 5. If they wanted help I was always able to be there to give it - even when I was working night shifts.
Not to mention music practice and this kind of thing.: See above, everything from here is a process of letting go of control - if they want to play an instrument they need to be self disciplined in their practice - If they don't want practice to then that's their decision. You're not superwoman, you don't need to control everything.
Finally, I would say that if your partner prefers you to be at home, and you don't really want to go back to work (as your OP implies) then I'm not sure why you are asking these questions? I suspect (and hope I'm wrong) that you were hoping for people to confirm that with no help and 4DC you couldn't possibly go back to work and 'do it all'. I really hope not, because that feels a bit insulting to those of us who have to work to support our families. The long and short is - if it's something you have to do you'll find a way to do it. If it's not, then you'll do what you already seem to be doing, and find as many possible excusesbarriers to going back to work as you can to justify why you 'can't' do it.