To ask how to run the house once back to work

I think those 3 days he needs to have tea on the table when you walk in the door at 5.30. Even if the dc eat at childcare they might want something? Otherwise you're well into overtired hungry kids crying.

I would put a wash on overnight and DH hangs it up in the morning. Then you do the rest of the washing on your days off / weekend.

I don't know about cleaning. I hate it, and we don't even have a cleaner any more, I do the absolute bare minimum!

Write a cleaning checklist. - lots of examples online and TOMM a great place to start - but crucially have a go then rewrite them to suit you.

Get a really good cleaning kit - I have two robot hoovers, one of them can mop if a bit half heartedly, a vileda spray mop, pledge fluffy duster and feather one, innumerable cloths and a little closed bucket with handle for chucking used ones in pre wash, a cleaning trug upstairs, cleaning spray and bleach in each loo, a small shark wand for doing the stairs… eyeing up a robot window cleaner for next year. Oh, and clothes brush in a handy drawer for wherever there’s a sofa. Brush dust crumbs and other debris off sofa on to floor before robot hoover starts.

i sound like a maniac (quite possibly am), but I just lower barriers to entry everywhere. Like at night, dinner cleared, hoover bumbling about on its own while I wipe surfaces and put dishwasher on etc.

2 kids, 75 million kids activities, volunteering, work full time, DH works away every week, no cleaner or domestic help - please be assured it’s frequently chaos and indeed upstairs bathrooms need seriously deep cleaned this weekend, but we get there.

Also agree with about poster re "doing his bit". That is slightly better way of phrasing it than him "helping" but he has to view the domestic stuff as mutual chores rather than him "helping you".

Re laundry, time how long it actually takes to process a whole load e.g. sort it, put in machine, hang it then put it away. It's not that long in total maybe 10-15 mins max. Totally manageable between you to do one load a day, try not to overthink it.
In your position I'd be increasing cleaner to once a week and trying to batch cook at weekends or get a stock of decent ready meals in e.g. cook if you can afford it.

In that case putting a load in the wash, a load in the dryer and folding and putting away a load seems like a good daily task for your DH when he gets in. He can do it with a cuppa.

Don’t fall into the trap of him doing’his bit’ - he’s a parent and should want to be doing an equal share of parenting his children and running the home.

Split the nursery run - one does drop off and the other collects. Diaries aligned for family events and children’s parties.
Split time off work to cover illnesses. Outsource as much as possible - online food shopping, buy presents online (which can be wrapped and delivered) and a cleaner as much as possible to put the shopping away when it arrives, change beds, laundry and get meals prepped so you only need to heat up.

We're old school, so have a list of all the jobs and tick them off. We do about half each. We do discuss quite a lot, if you're going to share the burden there has to be some chat.

If you do the pickups then he needs to use the hour that he has to at least cook dinner and do a load of laundry. Easily doable in less than an hour.

He has very soft hours and plenty of time to do his share and still give you both down time after work.

If he sorts dinner and laundry then you can do bathtime and homework and take turns for bedtime.

How old are the kids?

Pretty much what @exprecis said, though DS is old enough to get himself to and from school.

Our set up is similar in that one works part time and one full time but just longer hours. We share bedtimes - one person cooks, does dishwasher and tidies toys whilst the other does bath and bed. We switch this up as suits our schedule.
We also have very specific jobs which we each do that helps. I tend to use first thing in the morning to complete whatever is most pressing whilst he does breakfast and getting kids ready. I also do this on a Saturday as we are up early.
I have two set washing days which helps a lot with reducing clothing related mess

We both work full time, but this is how we do it.

Cleaner once a week, neither of us do any cleaning apart from that, spot cleaning or quick once over if we have guests over. Cleaner changes the beds and they go straight into the wash that day.

I do drop off, husband does pick up.

I cook dinner, husband does bath time.

Washing - either a load overnight and hung up in the morning, I do it on my lunch hour if wfh, or husband does it when he gets home. It takes less than 10mins to hang the washing up, so it’s not really a big issue.

Shopping is all online, cleaning products, pet food, etc is all on scheduled delivery so we don’t have to think about anything.

Weekdays I get home, we all play a game, have a chat, then bed/bath and I cook dinner for us (kid’s dinner is leftovers from our dinner last night, or something equally quick). We eat, chat, listen to music, watch tv, read, whatever - no housework.

Weekends we usually go out - again no housework.

I honestly don’t find it stressful and seem to have a lot of free time!

Out of interest why will you be doing all of the nursery runs when he has shorter hours?

What we find works best is each of us having our own patch rather than sharing jobs - except for school runs which we share equally.

For you guys - he has 4-6pm 3x a week child free, and 2x with the kids around. You have two days with the kids so you're home a bit more but with young children to look after.

So I would say to use that time effectively -

I would suggest he cooks dinner every week day, and also that he picks up the top up cleaning which he can do on the 3 days the kids aren't around - he should be able to hoover one day, do kitchen one day, do bathroom another day.

You could then pick up the washing on the two days you're around because that is easier to do with kids around than cooking or cleaning.

Food shopping - just get it delivered.

Practically meal planning, batch cooking in advance and freezing plus never letting things build up so it never gets unmanageable.

Or up your cleaner and get them to do washing as well?

We just started using an app called Nipto where you can add all the chores (it has loads that you can just select so you don't have to add them all) and then whoever does them collects points and so you can see how evenly stuff is distributed. Was useful to map out everything and discuss how much tasks are worth so invisible jobs become more noticeable and broken down into components. We both with FT though so equally sharing stuff seems obvious (though doesn't really work out in practice). I know you are doing childcare when home though perhaps doing the odd wash those days isn't so bad? Or does he think you should do everything then? Does your DH not want to do anything? Is that the problem? Or is it just working out a schedule? How many DC and what ages?