It is totally fine for you to stay in the UK past the 6 months on your US passport, since you have citizenship/right of abode.
However, if you leave the UK, and return again using your US passport, you will face questioning based on the assumption that you have previously overstayed. This can be cleared up by showing your UK passport/anything else official that identifies you as a citizen of the UK, and explaining what happened. You can also completely avoid this by entering the UK with your UK passport on any subsequent visits.
However, these types of things give me OCD-shivers, so if I were you I'd still do as the other poster said and visit a UK Immigration office and have this matter sorted out.
For your convenience, here is the contact info you need.