The part about the rood screen makes me wonder-- did children sneak pockets full of tops and pea-shooters into church? Did they get caught with them, and have to leave them behind? I am sure then as now some frustrated parent or teacher might have said 'give me that."
I wonder if people back then said 'You cannot bring it with you unless you have enough for everyone.' Or 'What have you got in your mouth?'
The paintings show a world where kids had lots of ideas of games to play, and some adult must have made the tops and balls. It makes me happy to think of children at play, then as now, watched over by adults with perhaps the same mixture of irritation, and fondness, with a little memory of the past thrown in.
As far as research-- not much help. I can give small Marlowe and Kyd tops, and knucklebones, dice, and balls. Archie and Horatio and Percy and Will might have blown soap bubbles, and had kites maybe. Anything vaguely the right shape can be a toy boat, (acorns.) And finding something to throw and then throwing it is classic. And I am quite sure, then as now, boys took sticks to play swords, or run around going 'bang, bang!' When they ran down a hill with arms out they were playing birds, not planes, but the difference is small.