@Anythingelseintheboxpandora
I think your expectations are askew. Maybe you had unrealistic hopes of an organised life when you moved into the nice new house and the resulting disappointment is all the more keen?
I doubt the DDs understand what you mean by a general suggestion to 'tidy up'. They appear not to have enough focus or engagement with the idea to keep at it, and not enough sense of being a team to help out. This is all completely normal behaviour for children aged 6 and 4.
It's unreasonable to set a rule that one thing gets put away before something else gets taken out. That's not how the flow of play works. Children often amalgamate materials from all sorts of different sets, and they drift from one set of toys to another according to varied scenarios of a game. This is also completely normal play behaviour.
You need to accept that if that's how they play, and not let it get to you.
I assume you are all out of the house all day if you're a working mum, so the messing happens when you're all at home together at the weekend and in the evening? As one suggestion, try to figure out how their nursery or CM (if they had one) kept on top of the toys, and copy that.
But this isn't just about tidying. It's about developing a co-operative spirit.
While the mess is frustrating to the point of pulling hair out, the tantrums and lack of co-operation are the real problem.
Wait the tantrum out. When the storm passes, ask the child about her feelings, whether you can help her express them in a way that is easier to understand than screaming and crying is. Listen to whatever answers you get. Offer a hug and say you are happy she is feeling more calm.
Tantrums can happen for a variety of reasons:
Child has used tantrums to get her way before.
Child is tired or hungry (hangry).
Child wants you to hear something she is feeling and doesn't know how to say it.
Maybe there needs to be some interest shown by you in the game they are playing, and an assurance that something they have built isn't going to be taken apart and thrown into boxes. It sounds as if they have built Lego furniture for the Sylvanian family?
Maybe there's stuff that isn't really being played with as it could be - Jenga is a game for older kids/family. If there are wooden blocks, pile the Jenga blocks in with them, but if not, maybe put that one away.
Go through toys with the girls and ask them if they play often with them and if they would be willing to create more space for playing with the toys they are really interested in by putting some other stuff away for a while. It's very important to involve the girls in decisions about toys to be rotated or they will feel powerless and defensive about their belongings, and you won't get co-operation. Don't be tempted to swoop in and make executive decisions.
Get a family calendar and let them help you develop a daily schedule that includes tidying time. Get them in the habit of doing things at the time on the calendar - lunch for instance, or going for a walk on Saturday morning. Or helping put away their laundry with you.
Ask them what time they would like to tidy together, and mark that activity on the calendar. When the time approaches, give them a heads up about at seven minutes, then five minutes away, that it's almost time, then another at two minutes, and again at one minute. This allows them time to transition from their game to the new activity. Transitions are hard for young children. Make it easier for them. Use a timer to ding or beep when the time comes. It's hard to argue with a timer.
The reason for their involvement in the calendar/schedule is to help them feel you're all on the same side, that this isn't a case of Mean, Boring Mummy appearing out of the blue, trying to ruin their fun.
When you're tidying up, put on some upbeat music you can all sing along to, maybe try to beat the clock. When the tidying is done, do a round of high fives and compliment them on co-operating. Try and keep the mood light as you tidy, and encourage them as you go.
Keep a basket in all the rooms of the house for toys that need to be brought back to the playroom at the end of each day.
Make sure there are enough containers in the playroom to put little bits and pieces into, and more importantly, to find things they need next time they want to play.
If you have one big bin for 'Barbie stuff', for instance, maybe consider a number of smaller bins instead, for shoes, clothes, vehicles, furniture, the dollies themselves. It's hard to find a little shoe in a big bin full of lots of other stuff, and maybe this is why they empty stuff all over the floor when playing.
Same goes for Lego and anything else that comes with lots of little bits - a box for each colour works better than a humongous bin.
This is not an overnight solution. It's a general one though, and it will encourage co-operation in other aspects of your lives. It also encourages feelings of trust between the girls and yourself, so that they feel you can set your feelings /frustrations aside and will listen to them.
It's easy to shut everyone up and tidy furiously, dump toys, make threats, but while that approach will achieve your aim, and fast, it will also damage the trust in the relationship. In the long run, trust is something you want to build up in a family, not tear down.
To start out, take time to do a big cleanup of everything, so they have a cleared space for play, and also so they see what you mean by 'tidy'. Don't do this in a spirit of anger. Don't fling their things around.
After that, you can develop the new system.