When you say 'admissions panel' do you mean an appeal panel, or just the admissions team at the LEA (who will have been the ones who made the decision, unless this is a foundation school)?
What is the Published Admissions Number for the school? If it is a multiple of 30, the LEA will be bound by the infant class size rules, which limit any class to 30 pupils. As your friend was a late applicant (not through any fault of hers, I know) the class was no doubt full by the time she applied. Her child cannot be admitted as the 31st pupil.
Your friend can appeal, in the hope of overturning the decision. But as others have said, the difficulty of walking to school with a toddler in tow isn't grounds for appeal. She would have to demonstrate that the refusal of a place was so unreasonable that the panel should overturn the decision and award a place. That is extremely unlikely to happen - especially if this is an infant class size appeal - but she may be lucky and find a sympathetic panel which is willing to stick its neck out.
Your friend needs to investigate other options. The choice adviser (or similar title) at the LEA should be able to provide information about any other schools with places. Through local toddler groups, can she find another parent to share the school journey with?
Where is she on the waiting list? Waiting lists are held in the same order as the admissions criteria, so if she lives very near the school she may be near the top of the list.