I have to produce ID for OU exams which has a photo and a signature (e.g. passport, driving licence). There are alternatives listed for people who don't have any of these - but it generally involves bringing something official with a a photo (such as a Citizen Card or a Validate UK card, which can be bought for just such purposes) plus a credit card for siganture verification. In the same place they also state that expired passports and driving licences can be used, so long as the photo still looks like you (fortunately for me, as I have an expired by date-on-the-front [but for complicated reasons, still valid to drive with] licence!)
I would very much hope that ABRSM have forethought enough to realise they will need to give alternatives to photo ID - what forms of photo ID do most children have? (I can't quite imagine my mother letting me take my passport in to school to use as ID even post-16!)