Flights were booked individually.
While UK immigration rules do not require you to have a visa for transit purposes (regardless airside or landside or if you have a self-connection or single itinerary) when you hold a common-format residence card from Germany, individually booked flights (self-connections) have their own challenges, especially with a low-cost carrier that often operates exclusively under a point-to-point model and do not sell connecting tickets in a single booking.
With self-connections, the airlines often do not recognize the validity of the onward ticket required to benefit from visa exemptions for transit reasons. With low-cost carriers, the ground staff is usually also much more conservative in the interpretation of immigration rules to avoid immigration-related fines by the government (even if they have to pay the EC261 compensation, which is comparatively small).
The airlines claiming that the refusal of boarding for visa-exempt self-connections is justified usually invoke two points. First, for self-connections, the airlines do not have a contractual obligation to transport the passenger to the final destination (the two tickets are separate contracts), thus, if delays or cancellation happen, the airline is unable to guarantee the prompt departure of the passenger within the transit rules. Second, many airlines do not have direct access to the bookings of airlines outside their alliance or their chosen distribution system (for low-cost carriers, they essentially do not have access at all). The airlines often claim that as a result they cannot confirm the validity of the onward ticket.
I do not believe these arguments are reasonable, at least in some cases (e.g. tickets booked with the same airlines), but what I believe does not matter.
I am aware that some civil aviation authorities (e.g. Denmark) also consider such denied boarding unjustified, at least when both trips are booked with the same airline (so they have the ability to confirm the bookings); however, civil aviation authorities do not have power to compel payment.
Their ruling is rather meaningless if the airlines does not comply (in the Danish case I'm aware of, RyanAir refused to comply with the ruling and pay the "ordered" compensation). Even though they usually have some authority to fine the airlines, it would require a formal proceeding that the authority may not be willing to commence for various reasons and the fine is paid to the state (not compensation to the passenger).
The passenger would have to rely on the civil procedure (e.g. small claims process or a binding arbitration for consumer claims) to obtain the compensation and the court may or may not rely on the ruling of civil aviation authorities depending on the country's system and the judge.
I am not aware any clarification from the ECJ or a national appeal-level court rulings on the denial of boarding due to visa reasons for self-connected flights.
You should still request compensation based on EU rules, but it may be an uphill battle and the airlines is likely to refuse. If you have the time/money and are willing to take the effort, you can attempt to obtain a ruling from the court (usually the small claims process); in some cases, some airlines may tend to settle to avoid a legal precedent.
Attempting to avail yourself of the transit visa exemptions as a holder of a less "favourable" passport remains risky when you do not have a single booking from the airlines or a proper travel agent (i.e., not Kiwi or other "self-connection" providers). I would advise against it when the trip is important enough.
I could only afford a transit flight through the UK (London).
Self-connected flights are often cheaper for many itineraries served by budget airlines. But being denied boarding and having to buy new flights or cancel the trip is more expensive.
