For a Oneworld Round the world trip, you can fly BA City Flyer because it is operated as a subsidiary of BA, just like Qantaslink for Qantas or American Eagle for AA. London City is much closer to the city centre than Heathrow and is much more convienient. The BA city flyer product is quite a bit better than their normal short haul product departing Heathrow. City flyer still serves drinks and snacks for free rather than an LCC style no frills experience on BA short haul. Do note though that for Business Class, no seats are blocked for the ‘european style’ business class.
I’ll also note that if you don’t land in LCY on your previous flight, departing from LCY on your next Cityflyer flight would constitute a service sector because you switched airports even if you are still in the same city.