Do I have to prove games consoles belong to my child?

Yes

Bailiffs are not interested in children's toys when they are recovering an unpaid debt.

They threaten to take things that mean a lot to the debtor or their children to spur them to pay the debt.

If bailiffs actually took control of a child's games console, the auction price would be negligible.

If the child, or person under 16, is the debtor named on the warrant, the bailiff cannot take control of the child's goods.^[1]

Otherwise, the child's parent or guardian can make a claim to controlled goods.^[2][3][4]

^{[1] Regulation 10 of the Taking Control of Goods Regulations 2013
[2] Paragraph 60 of Schedule 12 of the Tribunals, Courts and Enforcement Act 2007
[3] Part 85.4 of the Civil Procedure Rules
[4] See: Making a claim to controlled goods}