I sometimes take my grandchildren in black cabs on London visits as I can't cope with tube station stairs with baby, buggy and toddler, and busy buses won't let you on with buggy unless you fold it up, which generally requires removal of infant and rapid growth of two extra arms to hold all the stuff stored under buggy for the day.
At first I was worried about safety, and safety regs, but cab drivers assure me that children are allowed in black cabs without the usual restraints. I have got used to it, repeating to myself, "This vehicle is akin to a tank" and " How often is a taxi passenger injured in slow-moving London traffic?".
Baby and accoutrements stay in buggy, which is parked in the big space. Some newer cabs have straps for securing wheelchairs and they can be used on a buggy, but you still need to keep a hand/foot on it. I hold toddler on my knee, within my seatbelt.
Good luck Stilllgood, enjoy the Zoo!