My daughter was 4 when I went back to work full-time (not in daycare) after a year's mat leave. Although she had been in daycare prior to my mat leave, she had certainly become accustomed to being at home with me. I honestly think the adjustment was harder for her at age 4 then it was when she was 1 years old and just starting out. But it was only really bad for the first few days and then she was mostly just shy and unsure. Once she got to know the daycare provider and the other kids better, she settled right in. All in all, it was about 2 weeks of adjusting.
For my own daycare now, I just assume that most kids will need a couple weeks to adjust no matter how old they are.
I had a 4 yr old like that and it was behavioural and for attention. He'd be playing fine like yours then cry out when we had to clean up. I had to give him time outs. I said I understand U are sad and miss ur mommy. It's ok to miss her but screaming and crying like this is not ok. If u wish to do that then do it in time out. And he did. He needed a moment to himself to compose himself. I made sure to leave some toys out so once he was done he cleaned up and didn't pull a crying stunt to get out of cleaning. Basically u have to tell and teach him at age 4 that yes it's ok to be upset and miss mommy all u want but at 4 yrs old there are proper ways of showing it. Screaming and acting out is not acceptable and warrants a time out. It worked really well for my guy. After two wks he was done. He probably thought it's boring constantly being sent to time out so he stopped the screaming.
I'm okay with sadness, and being shy, but screaming and sobbing is the part that seems extreme for the age?
It may seem extreme but perhaps it isn't for this particular child. It may be that he has separation anxiety and that isn't something that is going to go away easily at age 4 and will take some additional work for you outside "waiting it out" and a regular adjustment period.
I advise that you work collaboratively with the parents and establish patterns of behaviour at home and how parents respond to certain things. This will give you a better idea of what negative behaviour is being reinforced rather than making assumptions and getting frustrated. Aside from the crying is this child well behaved? If so why not have a space where the child can be alone but in your sight, with comforting items as long as he is calm and quiet and then he can join the group when he is ready to join in and be happy. He is old enough that you should be able to communicate with him enough to find out what he needs from you to be happy in care. I imagine once he has settled in the younger sibling will follow suit. Have you got a play tent you can put up so he can go in there and play cars or read books alone till he cheers up. You are offering him a safe space that way where he is allowed to feel what he feels but at the same time letting him know that outside of that space he must not sob and cry as it is disruptive to the group. Whenever he feels sad he can go into the tent, or space, and then come out when he feels better. Eventually he will not need it anymore.
It kind of sounds like the child is overwhelmed by the sudden change in life. And, if his sibling is not coping well either then it would make his adjustment harder. At 4yrs old if you are stressed about something, having another child bawling and stressed about it doesn't help you build confidence!
I think that as hard as it is for a 12month old to adjust to daycare it can be even harder for an older child. 12 months old don't really have a clue whats going on and just get used to the new norm...4yr olds can sit there all day remembering what their life was and comparing and grieving the loss of old ways etc. While it is harder to have patience for them at that age we still need to be put in their shoes and figure out what they might be reacting too and try to work with it.
I'd suggest identifying the emotions. "it's hard to not be at home anymore, it's hard to not have mom with you all day" etc etc. Give warnings before you transition. Transitions are really hard on kids that are not settled and feeling safe and confident. Give lots of warnings as to what is to come so they aren't taken by surprise.