It seems like this is explained pretty clearly at https://tfl.gov.uk/fares-and-payments/travel-for-under-18s/zip-oyster-photocards#on-this-page-1. You would have to apply for an Oyster Zip Photocard online, several weeks in advance, and there is a £20 administration fee (£15 if you live outside the UK) which is not refundable. So for a one-day trip, it is probably better just to buy an ordinary adult Oyster card for the older child, and pay the adult fares.

Your younger child can travel for free:

Your child doesn't need a 5-10 Zip Oyster photocard: [...]

• If you accompany your child on Tube, DLR, London Overground, TfL Rail and some National Rail services. Check this map to see the services on which they can do this.

You can return the cards and get the deposits refunded at any ticket machine, as well as any remaining pay-as-you-go credit, up to £10 per card, See https://tfl.gov.uk/fares-and-payments/replacements-and-refunds/oyster-refunds-and-replacements

It seems like this is explained pretty clearly at https://tfl.gov.uk/fares-and-payments/travel-for-under-18s/zip-oyster-photocards#on-this-page-1. You would have to apply for an Oyster Zip Photocard online, allow a week or more for delivery, and there is a £15 administration fee which is not refundable. So for a one-day trip, it is probably better just to buy an ordinary adult Oyster card for the older child, and pay the adult fares.

Your younger child can travel for free:

Your child doesn't need a 5-10 Zip Oyster photocard: [...]

• If you accompany your child on Tube, DLR, London Overground, TfL Rail and some National Rail services. Check this map to see the services on which they can do this.

You can return the cards and get the deposits refunded at any ticket machine, as well as any remaining pay-as-you-go credit, up to £10 per card, See https://tfl.gov.uk/fares-and-payments/replacements-and-refunds/oyster-refunds-and-replacements