Maryzed I love your 'tissue' example. I can't stand 'tiss-you' or 'iss-you' (issue), as they sound so very self-consciously affected, your granny is right!
OP children are extremely adaptable and pick up and drop accents far more easily than adults. Fitting in at school is important. It's one thing to be one of a significant minority at school speaking in a neutral accent (hardly anyone actually speaks RP these days), quite another to be the only child to do so - not that they will, they'll just change back for you at home.
I can understand you not wanting them to be 'trapped' with a strong accent and no other way of expressing themselves but I think I'd be concentrating on vocabulary, grammar and clarity of expression. That way they will be able to express themselves clearly to anyone and you'll find that accent alters depending on situation and interlocutor.
It does bother me when people use regionally-varied grammar and colloquialisms, unusual to the extent that they obscure meaning, in public-facing work situations and I feel quite sorry for people 'trapped' in such accents to the extent that they cannot make themselves understood to people from other regions and especially to speakers of English as a foreign language.
Or, if it bothers you that much, maybe you should move to a more upmarket part of Birmingham or get your child into one of the grammar schools! There is definitely a 'neutrally accented' population in Birmingham, as in any large town. You could try engineering friendships rather than accent directly!
You're going to have to get used to the short vowels though. Are you aware that it is pretty much only people from SE England, in the whole of the UK, that use them? You're the oddities with the strong regional accent 
I am from much further north than B'ham and speak lovely 'standard northern' English. That is, no-one can tell where I'm from, other than 'not SE'. (That's apart from the ignorant twits who think everyone outside the SE has a cloth cap, outside toilet and strong regional accent, therefore I must either be southern or privately educated). I like lots of regional accents though and, so long as people can be understood, really don't see the problem.