The family is now forced to stay in Scotland for the foreseeable future as Arran - who underwent his second round of chemotherapy yesterday - is too sick to travel.
The family is asking for help from anyone travelling to Glasgow or Edinburgh to bring some of their belongings from their Cromwell home to them.
Ms Cochrane, who began her career as a nurse in Queenstown, said she was staggered by the response already.
"I’m really overwhelmed with how amazing everyone is being with helping us - I really wasn’t expecting all of this."
The family needed to stay in Scotland for at least the next six months, but were managing with the support of their extended family there.
They were hoping to get some of their sentimental items to their temporary base, along with Arran’s toys and artwork, family photos, clothes, books and shoes.
Friend and Cromwell colleague Juliet Towers has set up a Givealittle page to help the family - the page total was nearly $5000 last night.
She said Ms Cochrane had "the biggest heart of gold".
"She takes on so much for everybody - she puts so many of us to shame; she cares so much.
"I just thought, ‘Well, this is the time that we can all do something back for her’," she said.
Arran was taken to hospital last weekend, where his diagnosis was confirmed and he was immediately started on a platelets infusion.
Ms Towers said his leukaemia had a 90% survival rate, but came with a "very harsh" treatment routine.
Ms Cochrane said she was keen to use Arran’s story to raise awareness about blood and bone marrow donation.
"Every country has its own criteria and if everyone can just spread the word, some children could have their lives saved.
"The ward we are on has 26 rooms; that’s 26 children and families right now going through this hell, and that’s just the ones on this ward on this night," she said.
