Short answer: no CE or UKCA mark means a shop legally cannot sell the toy, no matter how good its condition. This isn't the shop being fussy — it's UK product safety law under the Toy (Safety) Regulations 2011.
The mark is usually printed or moulded directly onto the toy itself, or on a permanently attached label — not just on packaging you're likely to have thrown away. Check the underside of plastic toys, the base of soft toys' sewn-in labels, or the battery compartment of anything electronic.
CE marking is the older EU standard; UKCA is the UK's post-Brexit equivalent. The UK government now recognises CE marking indefinitely for toys sold in Great Britain, so a shop can accept either mark — but always check with the specific shop, since individual policies can still be stricter than the legal minimum.
Check the shops currently accepting toys before you set off.