One book I really enjoyed about Queen Elizabeth was The Little Princesses by their nanny, Marion Crawford. The book was a lovely tribute, a bit gushy in spots, but it offers a fascinating behind-the-scenes look at the Queen's childhood. Unfortunately, when the book was published, the royal family Was Not Pleased by such an invasion of privacy, and Crawford was completely cut off. If only they'd known the media scrutiny they'd be facing in the future!

Here is a link that might be useful: The Little Princesses

Liz, as Sheri says The Little Princesses will give you a good idea of how the to-be 'Royal children' Elizabeth and Margaret Rose's day-to-day lives were lived in the school-room (Crawfie was their Governess not their nanny). By today standards the book is rather 'twee' and the girls had young lives of such dullness that it seems surprising that it wasn't just Margaret that let her hair down when she grew up. At least Crawfie tried to liven things up with a trip on the Underground and visits to the zoo etc but their early education was of the minimum (girls, you see!)
There are loads of modern books about, who's authors have jumped on the band-wagon.
One by the BBC journalist Andrew Marr Diamond Queen and her People is taken from a TV series he did and The Final Curtsey by Margaret Rhodes is by Q E II's cousin so has a few different memories.
She was on BBC radio the other day.
I should point out I haven't read the last two books.

Here is a link that might be useful: Margaret Rhodes on 'Desert Island Discs'