BA and Qantas used to have an agreement (may of been a codeshare) on the kangaroo route until a few years ago, when Qantas setup a deal with Emirates. Not sure if that is still 'on' either though.
A correction on the fuel stop, all of BA's Sydney flights now stop (or did for the time being) at Singapore.
As an aside, a London to Australia flight non stop has been technically possible for years (B777-200LR), but the payload restrictions make it commercially unviable
BA and Qantas have a fairly unique relationship; BA used to have an equity stake in Qantas and, for a long time, merger/acquisition was actively discussed. The stake was disposed of some time ago.
The two carriers both remain members of the
oneworld alliance, and there are codeshares on certain routes.
However, and more recently, Qantas entered into its joint-venture with Emirates, and revenue-shares with them on set routes. Emirates is non-aligned (ie not part of any alliance) and IIRC, longhaul Qantas flights which revenue-share with Emirates do not offer codeshares with BA.
By contrast, Qantas domestics and Trans-Tasman routes don't revenue share with Emirates, but can and do operate as BA codeshares with BA flight numbers.
As an aside, the Emirates JV is one of the reasons that Qantas keeps announcing extensions to suspension of longhaul ops; the demand is more than catered for with Emirates aircraft operations, so they just draw the revenue from those instead.
As a further aside, and point of uniqueness, an Emirates-operated flight booked under a Qantas codeshare (ie ticketed on Qantas 081- stock) actually permits the accrual of Avios and Tier Points in the British Airways Executive Club. It's the only time that a flight operated by a non- or mis-aligned third party is eligible for Executive Club earning.
You're quite right that the only BA route to Australia for quite some years now has been the BA15/BA16 rotations through Singapore. Bangkok hasn't operated a continuation since Sydney went once daily, and Kuala Lumpur was suspended completely, let alone the continuation, for quite a number of years, too.
The BA15/BA16 has been commercially sound since the move to B777-300ER operations, helped in part by good basic operating costs and also the strong cargo demand for BA between Sydney and Singapore - and the B777's enormous payload beneath the floor.
You're also quite right that the -200LR can do Australia non-stop, and indeed a British Airways one has done London to Perth before, as part of a Commonwealth Conference in Perth, taking the Queen. As you say, the 777s op costs don't work for a commercial service, but that's where the 787 and A350 come in.
The Qantas London to Perth direct has been a success for them, and a lot of the behind-the-scenes work on 'Project Sunrise', which is the A350 associated plan for Sydney/Melbourne to New York and London, has actually continued apace during Covid. Test flights have operated, with crew and volunteers wired up to monitor them physiologically, and test different catering protocols, hydration effects etc.
Qantas apparently sees non-stop services aimed at the highest-yield customers as being potentially even more commercially viable post-Covid, as it takes out any risks associated with a transit.