I found the Elizabeth SMITH b 1846 Wavertree in both 1871 and 1881

1871 she is at 34 Stafford St Liverpool (RG10/3788 90 pg 14 #60)
Elizabeth SMITH Serv Unm 25 Domestic Servant born Lancashire Wavertree

1881 she is at 3 Buckingham Crescent, Rusholme, Lancs (RG11/3911/112 pg 7 #23)
Elisabeth [with an 's'] SMITH Serv Unm 35 Cook born Wavertree Lancashire

In 1861 there is an Elizabeth Smith age 21 Servant in Wavertree, born Wavertree. She should be 15, so that's a considerable gap. I couldn't find another one, though.

Our Elizabeth Smith married Joseph Scarffe in 1885 Douglas. I looked ahead to 1891 and 1901 to see if there was an unmarried Elizabeth Smith b Wavertree about 1846, but there was none. If there had been, we could eliminate this Elizabeth.

Every bit of evidence so far points to this being the correct Elizabeth Smith, other than there being no connection found to Lytham St. Anne's as Lynda Dickenson's mother-in-law remembers her mother (Florence Nightingale Scarffe) saying.

Incidentally, Elizabeth's likely grandparents Smith were on the 1851 census:
At Wavertree Hill, Wavertree HO107/2192 1059 pg 30 #117)
Thomas SMITH Head Mar 64 Gardener, born West Derby Lancashire
Jane SMITH Wife Mar 62 born Llanidloes Mongomery (Wales) .

Lynda, it may be worth getting the Dec 1841 marriage for John Smith that I referred to earlier. When ordering, put a constraint on that the wife must be "Mary Nightingale". If it's not, then one large bit of circumstantial evidence would have to be tossed out.