My daughter sat the musical aptitude test this year, scoring high enough to move through to the auditions. I mainly relied on the basic musicality she has but felt a bit of practice would not go amiss.
For the pitch part of the test (1/3 of the marks), we had a few goes of this just in the couple of months leading up to the test: http://tonometric.com/adaptivepitch/
IMO, this will MORE than prepare your child as my daughter said most of the pitches were much wider, apart from the last few. I have a higher degree in music and scored nowhere near 100%; don't be put off!
Assuming you have access to a piano, you could practise some of the texture questions (again, another 1/3 of the marks) by playing any combination of 2, 3 or 4 notes. I have NO idea how challenging the questions were on the day(some chords are easier to work out than others) but this would certainly give a helping hand. My daughter was getting about half right when we did them together. Quite tricky!
Good music lessons will prepare your child for the melody and rhythm parts so, although this hasn't answered your question about tutors, I hope it provides some ideas as to how you could have a little go at home, nearer the time.
All the best
(My first post after lurking for some (nail-biting) months)