Do airlines book you on other airlines if they cancel flights?

Yes, most of the times. In my experience 5 times they did (All international flights in the past 3 months), 1 time they didn’t (national flight within Argentina). Emirates and Qatar airlines did it automatically, Gol airlines (Brazil) I had to insist until they did it. I guess Lufthansa will do it without hesitation, but I could be wrong.
The time I went from Australia to Argentina, the final flight (Sao Paulo-Buenos Aires by Gol Airlines) was cancelled long before I took the flight (like a month before-I had the flight booked three months before). Sill when I arrived to Sao Paulo, Qatar gave me another flight with Aerolineas Argentinas and paid my hotel and food for two days in Sao Paulo.

No, usually they won’t do so willingly; however, with flights departing from an EU/Schengen state, or arriving in an EU/Schengen state on an EU/Schengen country’s carrier, they’re obliegd to per the EC261 regulation if no flights of their own are available and the passenger insists on re-routing rather than reimbursement.

Had this happen with a Eurowings flight PRN-DUS (as part of a PRN-DUS-MAN itinerary) which was cancelled 3 days before departure. Wrote them an email instantly telling them (in German) "this obligation applies regardless of any extraordinary circumstances; as such I strongly advise against trying to weasel your way out of it using the COVID-19 card, and I WILL drag you to court if necessary". 1 day before departure they e-mailed me with an Austrian+Lufthansa itinerary PRN-VIE-FRA-MAN.

In your case, as you’re flying from the UK, EC261 applies, so email or call the airline (or both) and tell them you’re insisting on a re-routing at the earliest opportunity as per Article 8 of the EC261 regulation (borrow my wording if you wish). If they still refuse, pay for a new ticket yourself and then request reimbursement. If they refuse, contact the CAA at [email protected] attaching all correspondence as evidence, and they’ll sort it out for you.

This is definitely possible, but I wouldn’t recommend expecting it.

Most major airlines have what are known as interline agreements which include provisions for transferring passengers in the case of irregular operations (i.e. flight delays or cancellations.) These are especially common for airlines within a given alliance, but also commonly exist among competing major carriers. For example, in the U.S., Delta, American, and United will transfer passengers onto each other’s flights when flights are getting cancelled, even though none of them are partners.

Lufthansa and Singapore are both members of Star Alliance, so it is somewhat likely that LH would be able to transfer you onto an SQ flight… if Singapore continues to operate those flights and neither the British nor Singaporean government prevents you from traveling. Currently, the British government’s page on travel to Singapore says, "those in Tier 4 areas in England will not be permitted to travel abroad apart from limited exceptions," and, "Short term visitors from anywhere in the world are not able to enter Singapore."

Checking Singapore Airlines’ website, it does appear that they are still (as of late Dec 21) selling tickets on the LHR-SIN route for Dec 29.

Do they sometimes do it? Yes. Can you count on it? No.

Unless there are specific regulations in some countries that require it (not the case in most major markets I know of), airlines don’t have an obligation to do so. So it’s purely a business decision.

If you are a valued customer (frequent flyer with status and/or travelling in premium classes and/or with an expensive fare), then they have a lot of incentive to do it, because they want you to be happy and fly again with them.

If they would have to pay substantial compensation (e.g. due to EC261) or assistance (hotels, meals, transport to/from the hotel…), they also have incentive to do so.

They will more easily reroute you through other flights of the same airline or group (e.g. the Lufthansa group also includes Swiss Air, so they could reroute you via Zurich if that made sense), then other airlines of the same alliance and other partner airlines, and probably only as a last resort completely different airlines, but trying to stay under late-arrival thresholds for higher compensation may affect the order.

Of course, there would need to be capacity available on the other flights, which would usually be an issue especially at busy times like holidays. Remember that the “target” airline lay as well have their own requirements to reroute their own passengers, so even if you are an elite frequent flyer flying first class full fare, they may not have any option available.

It depends on the airlines and the local market.

1. I was once stuck in Denver and saw United customer rep rebook people on Delta, American and even phoning a Southwest agent.
2. I was once stuck in Toronto and an Air Canada agent wouldn’t even rebook me on UA (a *A partner).
3. I once missed a connection in Munich and Lufthansa rebooked me on a later Czech Airlines flight (all Lufthansa flights to Prague had departed) even though the misconnect was weather-related and CSA is not a *A partner.

Happened to me in Feb 2018 when my direct Virgin Atlantic flight from Washington DC (IAD) to London Heathrow (LHR) was cancelled.

-They rebooked me on United (also direct) leaving next day.

-I called them up to see if they had any option for the same day so that could save me a day which is when they rebooked me from United to Delta (via Detroit).

-But then I showed up at the IAD airport on the day and time my original Virgin flight was scheduled and asked if there was any other option, so they rebooked me on British which was scheduled to leave in 2 hours.

-All of these changes without any additional charges.

Do airlines book you on other airlines if they cancel flights?

Yes, that happens from time to time. It happened to me last year. Because of a flight cancellation from Munich to Newark, Lufthansa booked me on a KLM flight from Munich to JFK via Amsterdam. They actually offered a few options, but I chose the one with the change in destination because I live rather closer to JFK than to Newark.

can I expect them to rebook me onto a direct flight e.g. with British Airways or Singapore Airlines or some other route if such a possibility exists?

I don’t know whether you can expect that. It’s very difficult to predict what may or may not happen in these unusual times.