TfL issue the "Big Red Book" to bus drivers. This sets out what is expected of them in a range of situations including bus breakdowns / curtailments.
Big Red Book download pdf
See page 31 of the document / page 16 of the pdf which is where the quote below is from.
3. Issue a transfer voucher to each passenger who used Oyster pay as you go or a contactless payment card and needs to transfer to another bus. Remind those passengers not to touch in on the next bus.
Although I've never had to request a transfer ticket I agree with the various comments that real life experience is highly variable. Some drivers are excellent, others deny they exist and I've seen passengers have to go into a complete strop in order to get one issued. That really should not be necessary.
In terms of curtailing buses there are a load of reasons why it can happen - accidents, road works, delays to the service, need to get the driver off duty for a break or to remain within their hours, planned events, emergencies.
Ordinarily it is the operator's "I-Bus" room and the route controller that will intervene. All of the instructions are tracked via the I-Bus system. Drivers should change the vehicle status when they go oos due to a curtailment. This removes the bus from displays to the passengers at stops / on their phones. It also shows the bus is not in service.
TfL use I-Bus data to determine the payments to operators. Previously it was done on a sample basis with people standing on street corners at defined points watching what was happening and entering each bus that went past and at what time. I-Bus gives a close to 100% sample over multiple points along every route every day of the week and at night. Clearly there are all sorts of things that can go wrong with the data, equipment etc so no one is saying it is perfect. I expect operators challenge the calculations and will provide evidence, for example, if a bus has run but its I-Bus kit is defective. It is not unusual for buses to "disappear" if drivers haven't logged on properly or the ETM goes wrong.
In certain circumstances Centrecomm (TfL's network control centre) can override the operator's control room and instruct buses to divert or curtail. This will usually be because of a major incident or severe accident where they can see via CCTV what is going on or the emergency services have asked that buses don't come into an area.