jenwa, the next time this happens, could you say to your MIL that there is one thing that you both agree on, that if your dd sees your mil, she is happy to wave but beyond that upsets her. And whilst you are sure (!!!) that mil does not mean to upset your dd, that is what seems to happen when she goes over to her - and obviously (!!!) you do not want to upset her again...
Say that just for the next time, the nursery staff have suggested that they will have the boys ready and that she can wave if she happens to see dd but that they want to see what will happen for 1 week if she doesn't go over to her. Point out that she is the adult and regardless of what she thinks (those auditory hallucinations of your dd calling her over! ) she is just to wave, that dd has been briefed and knows that granny is going to wave but not talk to her or call to her and she is not going to call to granny, just wave too and she is looking forward to trying this.
It also helps to get the nursery staff involved and thus lessen you as the baddy DIL, sounds like they are on your side, as it can't be nice for them to have to calm down your dd and the stress that it causes her and others in her room.
Make it just a one week experiment to start with, and maybe if you can bear it, sweeten it with a 'and if you DON'T upset DD, we can come and see you at the end of the day/some other time as a special treat.' BUt if she does go in then you won't go to see her when she wants you to.
HOpefully you will be able to make it sound better than I can as it is late and my brain isn't working properly!
Hopefully it will all go swimmingly well (or at least a bit better!) and you can then say that (through gritted teeth if necessary!) 'didn't WE do well at making sure DD was not upset when you picked up boys at nursery. Would be much better for everybody if we could continue like this'.
How do you get one with your sil? Would she be able to help by saying that the boys are upset by their granny going over to your dd and upsetting her/that they are in on plan and are going to be all ready for her to pick up/??? whatever so that she can't pretend that she heard dd calling/boy needs to pop to loo etc.
And Will nursery be involved enough to say that they will actively stop her from going over to see her when she gets there - even just 'reminding' her that 'this week we are ALL trying plan to make sure that dd doesn't get upset - remember just wave, boys don't need loo yadda yadda - pre-empting everything she might come up with to get to your dd...
good luck!