Since it has now been established that the daughter of Nathan and Elizabeth Brown was named Lily Clara Kate it appears that she was not the plain Lily Brown who married Walter Markwell!!
Lily C K Brown married a Robert Sharman in 1921 - she died as Lily Clara K. Sharman born 19/1/1896 in Sept.1970 Ipswich.
As has already been suggested you need to get the marriage certificate of Walter and Lily to establish the name of her father.
The surname Brown is always a minefield to deal with when little is known - indeed all we know (apart from her marriage) is that when she died in June qtr.1922 she was 26 I.e. born latter half of 1895 or early 1896. At her marriage and death she is plain Lily and there is only one plain Lily Brown born 1895 to 1896 in Suffolk - however, this is not 'your' Lily either as this one died unmarried in 1947.
So, either she was not born in Suffolk at all, or if she was hadn't been born as a plain Lily. There were only 3 Lily's born in Suffolk 1895-1896 - plain Lily above who died unmarried 1947, Lily Clara Kate who married a Robert Sherman and Lily Florence reg'd June qtr.1896 Ipswich. The latter had sisters who also had 2 christian names which they used as adults - however, at the moment cannot see what became of Lily Florence after 1911 so maybe this was 'your' Lily and she dropped her second name. We could conjecture till the cows come home.
The only way to try and identify who 'your' Lily was is to obtain her marriage certificate I'm afraid.
Annette