Sam, for my work I have had to study phonetics and pronunciation, and cab promise you that if yu want to learn the TH sound then of course you can, as long as your tongue can reach to your top teeth, then you can make the sound. If you really want me to I could send you diagrams about where your lips and tongue should be, complete with positions etc, but I am sure you could figure it out...
At music college, we had an old speech study lady who hated me because I have an accent. She wanted us all to adopt a southern counties accent, wasn't gonna do that, but I learnt to be able to speak properly if I had to. But an accent defines you, where you come from so don't worry about it. However the inabilty to make certain sounds of speech can make yu sound a bit simple, so you should learn to make a TH sound. I had to learn to roll my Rs properly, hard but achievable. So your Mum has so sense in what she says. But I am sure she is still a silly bitch who doesn't understand you etc...