Alex, The Leukodystrophy Charity (Alex TLC) supports families with all forms of leukodystrophies, of which there are more than 100 types.
Karen Harrison, its director of support, says only MLD and adrenoleukodystrophy (ALD) are treatable.
"Treatment is expensive, but so is the financial cost of caring for these children for the remainder of their lives," she says.
"We're talking about young children losing their lives in horrible ways, experiencing some of the worst symptoms imaginable while they are alive."
Josh Dean, the Labour MP for Hertford and Stortford, has raised Lily's case in parliament.
"I would urge the government, the NHS and the UK National Screening Committee to seriously consider adding MLD to new-born screening", he says.
"This will do so much to reduce the chances of this happening to other young babies and children"
The National Screening Committee is in the process of reviewing the evidence for a screening programme.
In October, the government launched a study to test over 100,000 newborns for more than 200 conditions, including MLD.
The findings "will help improve testing and treatment and help predict, diagnose and treat future illnesses," the government says.