You are right Slightly, consistency is the key here. If one adult says no and the other is getting all emotional and makes excuses, you will have a confused child. MIL needs to play by your rules. Firm, clear boundaries and all. And yes, following through the consequences, if you say you will be going home if she has a wobbly, then you go...no holds barred, no second or third chances. You did the right thing.
I totally understand your worries about nursery. If I were you, I would go and speak to them before she starts...tell them about the behaviour you have experienced and together, you can devise a joint plan right from the start. Star charts, whatever. Then you and the nursery can be using the same strategies so your DD gets a totally consistent message. I think it is going to be important for the nursery to be prepared so they can be proactive rather than reactive. They can keep a diary too.
What you say about your daughter's frustration /anger is interesting...she seems to get frustrated with herself doesn't she? Maybe play therapy could be helpful here? The nursery may be able to point you in the direction of a qualified play therapist. Might help get to the heart of the issue and give her a safe environment to express her feelings.
There will be a pattern to the behaviour, I feel sure of it. Just remember to praise your DD every time she reacts in a positive / neutral way, otherwise there is always a risk that we inadvertently reinforce bad behaviour by giving it loads of attention.
Remove from activity, put in car, silence all the way home, time out on arrival at home. And certainly no wailing from mother-in-law!!