I appreciate what you say but if you did not have leave to remain in the UK then you were technically illegally in the UK during this time.
The difficulty in terms of apply for naturalisation is that this means that during this time you would not be regarded as meeting the residential requirements.
Please refer to the following link and in particular the heading titled "Residential requirements":
You will note that one of the bullet points states "not been in breach of the Immigration Rules at any stage during the five-year period."
Not having leave to remain is regarded as being in breach of the immigration rules.
So, what this means is that the earliest that the 5 year naturalisation period could run from is from when you regularised your stay and were granted ILR in June 2011 (if you were not granted normal leave to remain between July 2010 and June 2011) . This means that the 5 year period can run from the date you received ILR.
This means that the earliest that you could apply for naturalisation is from June 2016.
I am sorry, but it's better that you know this so that you do not waste the naturalisation fee.
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