my youngest has raced the scottish cross season this year and really wants a cross bike for her birthday in a couple of weeks.
Now, she will be 7 and rides a 20″ wheel MTB.
Are there cross bikes that small?
And is there a second hand market for them?

I’m struggling to find any, except the expensive, new, 24″ wheel islas

https://www.frogbikes.com/lightweight-kids-bikes/road-bikes/frog-road-58.aspx

nice, but I can’t splash £400 on something that might be outgrown in a year.
anyone got a used Frog?

Riders of an age or height that are on 24-26 size bikes tend to only just at the size where they can operate STIs, even the small Microsoft versions. May be worth a try before you buy.

Second hand rigid 20″ mtb and put drop bars on? Would that even work?

Buy a used 24″ mtb such as a Kona Hula, swap the bars and shifters for road, old Sora with side buttons works for small hands, then change the tyres.

http://www.ebay.co.uk/itm/Kona-Jake-2-4-24-Kids-road-Cyclocross-Bike-/221666299078?pt=UK_Bikes_GL&hash=item339c574cc6