I don't think 3 years is too late. My dd will be 3 in Sept and she is down to attend a pre-school for 3 mornings a week from then. She attends a 'day care' type nursery one day a week when I work from 8 - 6 at the moment. Your childminder sounds great and I would keep her there.
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Yes she would be entitled if the nursery is part of the scheme. You would have to ask, as some private places ARE NOT.
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YOu'll get a lot of pros and cons, and need to go with your gut feeling. I did the run around last year at this time to put dd's name down, intending to start last Jan when she was only 2.3 but as that time grew nearer, realized that I was happier waiting until she was older (3). We go to toddler groups regularly, swimming, lots of meetups etc, so I don't feel she is missing out on anything in terms of socializing with others her own age at this point.
I am in the minority in the group of mums and toddlers I know - most have send their child to one of the groups you mentioned from the age of 2 or whenever they were 'accepted ' - some places won't start them unless they are dry, or until a certain age. I struggled with this one for a long time, and know I made the right decision for dd and myself.