Your simplest option in my opinion would be the Santander bikes which are all over the place, you have to pick one up and return it to another dock within about 30 mins and theres a cap of about £2 a day as long as each ride is under 30 minutes.
https://tfl.gov.uk/modes/cycling/santander-cycles
Lime bikes have also come to the uk, I believe you unlock with the app, and just leave them anywhere within the zone.
No idea about child attachments though, you'd probably have to bring one with you. As an aside, for the most part (apart from VERY few specific cycle only lanes) London is a VERY unfriendly city towards bikes, and if its your first time here with the traffic on the other side, AND with a baby, you'd have to be TOTALLY insane to ride a bike.
To emphasise that point an emergency room doctor I knew in London said they dont call them cyclists, they just call them donors....
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