I'm lucky all of the kids instincts were to call me Mum. At first whenever it happened they would look like they expected to get into trouble for it which isn't fair for anyone. My eldest felt it would be a betrayal without permission from her mother to call me Mum which is understandable.
My wife and I talked about it and naturally she didn't want me to have that title (which is fair) but we both agreed it can't be about what her or I want our first concern is the kids.
From their position and others it's pretty simple to them
Parent + Female = Mum
Neither of us wanted the kids beating themselves up and having to censor their own thoughts before they speak. We just want our kids to do what comes naturally and be happy like any parents.
After her asking our eldest daughter (12) if she wanted to call me Mum she answered yes. It was very brave of her as my wife's tone while asking the question made it clear she wasn't happy. To my DD honesty is very important even if it hurts and she laid it all out that they all had been calling me Mum by accident but they didn't want to hurt her.
It's sweet that they were trying to spare her feelings but they were hurting themselves in the process.
Neither of us could allow our children to that to do that to themselves and most importantly my wife didn't want kids going through that for her sake.
After that we made it official by telling the kids it's ok to call me Mum or any other equivalent.
In situations where it's needed to separate us she is Mum 1 and I'm Mum 2 which is cute.
We now have extra fun in our house from time to time when the kids just yell for Mum my wife will smile and say "They're shouting on you".
They can and do call me by name as well but I always light up when they call me Mum.
As to the loss of their Dad I'm happy to say there is no loss as they are all very happy with having 2 Mums