I too had this poem but in a maroon book with embossed front in my childhood in the late 40s, early 50s. I am sure my memory is right but I can only remember the first part. The "Nanny" referred to was, I thought, a large dog, but my mind was influenced by a sketch of a large dog next to Timothy. The poem went like this:-
Timothy Thomas was sick of the mumps
Timothy Thomas was down in the dumps
His face looked just like a full, full moon
Though they told him the swelling would go down soon.
His throat felt sore
He was sick and hot
And he hated the sight of his little white cot.
(Forgotten next line)
And as for Nanny, in quarantine
Whatever a word like that may mean
I can only remember one more line "his eyes lit up and his mouth went round" when he sees the 'Christmas tree'
Good luck, I would love to hear that you have succeeded in your quest, so that I might read this wonderful children's poem again.