No Olivia...because she's speaking British English when she's saying MUM, and you're speaking American English by saying Mom, she'll absolutely know the difference and she'll learn by it.
I have a friend in Spain, he's Bulgarian, he has never set foot in an English speaking country, yet he speak English fluently, as well as his own native tongue..and Turkish, and Russian, and Italian ( he works for an Italian company in Spain) Romanian, and Spanish like a native. he was completely self taught back in his native Bulgaria.. incredible..
I also know a Young (early 20's) chap who was one of my staff until I retired this year... I thought he was American when I first interviewed him for the position, his accent is generic American and fluent ... although he does speak somewhat like Steven hawking ( almost robot like)...anyway I thought no more of it until a few months later when we got chatting about his background... and in fact he is Bulgarian , had only been living in England for 9 months with his parents..had never set foot in an English speaking country until he came here to London... I was stunned, I asked him how come he had an American accent, and he said.. '' I learned the English language from watching American TV shows''..he went on to say that he hadn't realised that British English and American English were different until he arrived in the UK, and everyone kept asking him where in the States he was from...