I may be wrong but I do not think those sort of records exist, if they do they are most likely to be held by the local archive service. By the time of the 1939 Register my mother had already been evacuated from Leeds along with about 40,000 other children, she was sent just over 17 miles away. What I don't know is how long she was away or if her parents ever visited.
I don't know if this would be the contingent that your mother was in, but a history of Ilkley Grammar School ("Olicana's Children" by Peter Wood, 2009) notes that Leeds Modern School came as guests of Ilkley GS. To accommodate everyone, Leeds children were taught in the mornings and Ilkley ones in the afternoons. This continued until 1941, when the Leeds children returned home.
A number of my relatives were teachers at that time (my grandfather at Ilkley GS), and my impression from reading about those days was that often whole schools were evacuated together and education simply continued in a different place, so it's possible that school and education authority records might have something useful.
However, I think some evacuations were probably more random and haphazard: my grandparents took in a young cousin from Hull (like Leeds, a danger zone), and as far as I know this was a private family arrangement.