Nell is a traditional nickname for Helen. It's from the expression "mine Helen" -> "mine Hel" -> "my Nell" -> Nell. Same reason Ned is a nn of Edward: "mine Edward" eventually became "Ned". That being said, "mine Elizabeth" could come out as "my Nell" too by that logic... not traditional, but not illogical!
Nina is a generic nn for names ending in "ina", e.g. Catherina.
Elizabeth has an enormous number of traditional nns: Elisa, Tetty, Lizzy, Lizzie, Lizette, Lizbeth, Liza, Liz, Lise, Lisa, Lillian, Lilibeth, Lilibet, Lilian, Liddy, Libby, Elyse, Elsie, Elsa, Elly, Ellie, Ella, Eliza, Buffy, Betty, Bettie, Bette, Betsy, Beth, Bessie, Bess, Elle (these were straight from behindthename.com and they're quite exhaustive)
...If you don't like any of those, then Nell might be better to go with.