I can say no better than The Ghost of Christmas Past except to say this.
There is a precedent. Other than naturalisation, you can be British by birth or by descent, and you are British by descent if you were born outside the UK but have a British-born parent. When so many British colonies were becoming independent 50-60 years ago, the UK Parliament had to pass an Independence Act each time to define who will be a British subject (as it was called then) after independence day. The way it was done was to state that if you become a citizen of the new country under its new constitution, you cease to be British on independence day unless you or your father were born in the UK (because at that time only descent from a British-born father married to your mother counted - these days either parent counts), in which case you are British by birth or descent and can be a dual citizen if the new country allows it. If you have no close family connection with the UK itself, why should you continue to be British?
A few people "fell through the gaps" and there is provision for them to be allowed a British passport, but for the most part it worked. The main problem was with India, because it doesn't allow dual citizenship, and there are still a few (mostly very old now) people around who didn't fit being British or Indian and have been allowed to remain as a "British subject".
The logical thing to do is the same thing if Scotland goes independent, and having read all that I'm sure you can work out where you stand but here goes.... you were born in Scotland. If you have at least one parent born in the UK outside Scotland, you will be dual UK/Scottish. If both your parents were born in Scotland too, then you are thoroughly Scottish and after independence you have no claim to a British passport.
But this is all guesswork based on what has been done before, and the SNP hasn't even talked to the UK government about it. They NEED to. So for now, nobody knows for certain how it will be.