Hi Lina! Welcome to the forum.
Faerybee has given you lots of great information and resources to look through.
I can tell you honestly that when I first joined the forum, the ideas I had about bird keeping was outdated. And if I had bought two budgies at different times, before joining the forum, I would have put them in the same cage as well- because I didn't know any better.
And like you and your Mum, my Dad had done things the old fashioned way all his life (it wasn't old fashioned back then, it was the only way they knew with lack of research to tell them any differently).
Thankfully, while I still just had my first bird, I joined this forum and quickly learnt that even though some people can get lucky and *never* have any issues with introducing two birds, the stories where it goes wrong are always harrowing- for both owner and pet- and nearly always start with the phrase 'If only I knew how important quarantine was'.
So, while you have your two birds in together, it isn't too late too split them up and see through quarantine for the required time.
It is always best to have a cage as back up as a hospital cage any way, so having to get another cage isn't a terrible thing.
As your vet has little bird knowledge themselves, it might be worth enquiring if they know of any other practices that do have an avian specialist. While it might be too far for you to travel too, if they were able to speak to a colleague on the phone in an emergency situation, it would be very reassuring for you to have that back up.