If on a through ticket, you are protected end to end, your contract is to AMS, the airline will have to get you there. If you are 3 hours late then BA will just put you on the next available flight. If a BA ticket they would rebook you via LHR rather than LCY. For technical reasons that will be incredibly difficult to do on a 001 ticket mastered on SABRE but at least on paper they would do it if the computers were set up better. So probably they would instead rebook on the next LCY service, including other airlines. The London- AMS sector is also protected by EC261 so it's the law, with various remedies on things like right to care. If the ORD-LHR sector is on BA metal then that EC261 care is end to end too, so there is no option to BA, they have to help. Taxi getting stuck in traffic is why I wouldn't use a taxi in that circumstance (I'd probably Elizabeth line to Stratford or Canary Wharf and taxi/uber just the last bit) and that would be a grey area since the usual expectation, in London of all places, is that you use public transport. That would be protected from failure.

If you split the trave over 2 bookings then where BA metal you get EC261 protections on each section, but it's not end to end. Miss the LCY service then you have to buy a walk up fare and any return sector gets cancelled too.