It depends on the terms of the competition.
It may have been a term that by entering, you were agreeing to give all rights in the design to the school. You would presumably have had to read the rules and criteria in order to do whatever was required to be done and a positive action (entering the competition) can be seen to be agreement to those rules.
If there wasn’t anything in the competition which said that the intellectual property/copyright belonged to the competition organiser for them to do as they wished, then the copyright and any royalties in respect of use of the design belongs to you.
Copyright and yours for 70 years after your death, for the benefit of you during your life and your estate after your death.
I would be extremely surprised if it wasn’t a term of the competition that they had the rights to any works presented. However, if they didn’t, it’s an oversight on their part and you have the right be paid for whatever money they have made from it. If it’s a school, it’s going to be extremely difficult to quantify any profit they have made just from the logo. The measure of your loss would probably be how much it would cost them to have a logo made on a commercial basis.
Without seeing the rules of the competition that you entered earlier, it’s impossible to advise you further. You may not necessarily have signed anything but may have agreed to the competition rules by the action of entering
Can I clarify anything else for you?
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In the absence of any documentation, this matter would be decided on the balance of probabilities and the balance of probabilities afraid is likely to be that if they were having a competition, that the logo created would belong to the school otherwise there would be no point in them having it done.
If they bought a logo commercially, which might cost a few hundred pounds, they would be able to use that as they wished and hence, whoever created the logo wouldn’t be entitled to any of the profit on whatever they create.
By all means ask the school for a copy of the competition rules but don’t expect them to have them after 20 years. I think it’s unlikely that any claim would succeed if it got as far as court. Sorry.