Elizabeth is the most classic/timeless girls name in England (not sure about Wales, Scotland and N. Ireland) - as far as I can tell it hasn't left the top 100 since they started keeping written records in the 1300's! She'll have some amazing historical namesakes...
Yes, this means there are a fair few about of all ages - but it's still not as well used as the boys timeless ones (Thomas, James, William etc), so even once she's working age and mixing more with other generations it would be unusual for her to meet that many, let alone when she's school age. Plus of course many will be known be a nn or variation of Elizabeth - it's the name that has the most choices (over 2 dozen when I last counted!).
I think when the Queen dies it might rise slightly in popularity for a couple of years - but I would be extremely surprised if it reaches top 20. It's length and style means it shouldn't get very popular as the fashionable girls names these days most commonly end in an "ah" or "ee" sound and/or are vowel heavy and often the short two syllable type as well (Mia, Maya, Molly, Millie, Mila, Isla, Ivy, Ava, Ayla, Ada, Lily, Lyra, Lyla, Luna, Lola, Eva, Evie, Elsie, Ellie, Ella...).