Just bought a road bike for my 14 year daughter - I paid £350 new, I don't know if you will get much cheaper, however I suspect there would plenty of options on the second hand market. She is around the same size as your niece and she was good with a 45cm frome.
There is a triban 3 in the for sale forum for £150, nothing fancy but they get quite good reviews, shpock seems quite good for used bikes, i picked a giant scr up for £90 and a BMC pure pro with shimano 105 for £120. If your patient something will come up
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