From my own experience of watching many a walkabout, every gift / bunch of flowers received on a walk-about is passed by the Royal immediately to a lady in waiting...equerry...private secretary....and then back to a Royal Car. I have even watched a mayor 'pressed into service' when Princess Diana was on a 'walkabout' and her own staff had temporarily 'run out of hands'.
From my own experience, most people do in fact bring flowers - either in bouquets or singly. I have however seen people hand over photos [sometimes framed] of either the 'visiting royal' or his or her family taken on another occasion. I have seen people hand over teddy bears and other small toys. Sometimes, I have seen children handing over drawings etc.
I am hardly a 'scientific sample', but I have never seen anything 'valuable' handed over.
Flowers are almost always delivered later to hospitals, hospices etc although occasionally, royals might keep a token sample. [ There was a story in one of the better quality newspapers a few years ago that the Queen was presented with a 'home made' bouquet of wild flowers by a young fan, and the queen was so enchanted with the arrangement that she took it back to BP and it was placed on her dressing table. I hope the story was true, because it shows a nice touch on the monarch's part.]
Records are NOT kept of flowers handed over, but anything more sutbstantial is noted as a gift. Sometimes, the giver does note his or her name on the gift, and will then receive an appropriate thank you letter from a lady-in-waiting. If the identity of the donor is not known, then the gift is noted with the identity of the donor being stated as 'unknown'.
In one of the tv documentaries of the monarchy, I can remember an elderly man giving the queen photos that he had taken of her at a previous engagement. HM then passed these back to an aide. IIRC, the aide remarked 'how generous people had been'. From memory, nothing of great value was handed over however.
If the outing is to (say) an agricultural show or a flower show [where there will often be trade stands] the stall holder might well hand over to the visiting royal a sample of his/her goods/ produce etc. These tend to be fairly modest [small garden trug etc]. We used to be told that this practice was frowned upon, but provided the donor does not try to make commercial captial out of it, the gift is usuallly accepted and an appropriate letter of thanks is later sent by a lady in waiting. If the royal outing is to a charity bring-or-buy sale, if the royal sees something he or she wishes, the form is that the stall holder does not give the item free of charge, but presents it to the royal, with a lady-in-waiting etc then producing the money. I saw the Duchess of Kent do this some years ago when she was at a Charity Fair in London - after opening the Fair, the Duchess did a 'walkabout' of the stalls, spotted something she liked and the lady-in-waiting paid.
Following these principles, if a piece of jewellery were given on a walkabout, it would be passed to a royal vehicle and on return to BP etc its receipt would be noted carefully, with donor's name if known. Having said that, I have NEVER seen anything that looks valuable presented on a walkabout.
I hope this helps,
Alex