Does anyone have similar experiences of having three languages from start? What worked? What didn't?
I am French, my wife is Indonesian, and we talk to each other in English, so this is basically the same situation.
Our thought so far is to speak English when we are all together, Swedish between me and the baby, Italian between my wife and the baby.
That's what generally recommended, and that's what we did when we had our first child. I worked pretty fine for 2.5 years: The boy answered to me in French, and to his mum in Indonesian. It was lovely. He switched between languages without trouble and without mixing them. His mum would tell him "Ayo, pakai sepatu, kita mau ke kota", and he could fetch me saying "Papa, tu viens, on va en ville!"
However, when he started going to school (we live in France), he had more and more interactions with French-speaking people (friends, teachers,nanny...), bathed in a French environment, developed his new language skills in French only, and soon he started answering to his mum in French even though she talked to him in Indonesian.
This was a bit disheartening for my wife, the only one to use her mother tongue in the house. Also,holding a conversation in two different language is probably less natural for her adult brain than from our child's, and after a few months, she gave up and started talking to our children (the second one was still a toddler then) in French as well.
Nowadays the kids are 8 and 6, they speak only French although they understand some words of Indonesian and English. We feel like we failed at some point to transmit their mum's language, but we are unsure what we should have done better.
Maybe we should have been more stubborn and my wife should have kept talking to them in Indonesian. With retrospect, I think we should have set up some rules about the whole family (including myself) speaking Indonesian regularly, for instance everyday during dinner, or each Saturday, so that they could see that I was doing efforts too and that Indonesian was not only "mum's tongue".
I hope you can profit somehow from our far-from-perfect experience... Until 2yo at least, I think we had (and you have) the right plan, but you will probably have to device something more clever than us afterwards.