Certainly BL could have been as big as Toyota or VW are today, it did have many strengths when created, Jaguar, MG, Rover, Triumph and the truck and bus business were among the most succesful and well regarded in their markets. The problems lay almost entirely with the Austin-Morris volume car unit and this was what ultimately sank the ship.
I've been thinking about this over the last few days and based on AR Online's archive this is the scenario I've come up, I'd be interested in people's opinions, good or bad!
In 1962 Leonard Lord steps down as BMC boss to be succeeded by the aforementioned Joe Edwards. He's well regarded by the workforce and although BMC is still plagued with strikes as per all of British industry at this time, he manages to implement some changes in working practices. BMC continues to lose money due to strikes and warranty claims, the takeover of Jaguar to form BMH goes as per OTL but by 1966 the company is in serious trouble, again as OTL Roy Haynes is recruited from Ford and tasked to design a new range of cars, the first product will be the Maxi, for cost reasons the decision to use the Landcrab's centre section is still made but Haynes persuades Edwards to let him design all new body panels for the car hiding its origins, the Aquilla concept is too radical for this time and is probably not capable of being mass produced but it shows what could have been done with the Maxi chassis. If the Automotive Gods are smiling, the E-Series prototypes suffer crankcase failures in testing due to vibrations from the gearbox in the sump necessitating a redesign into an end on arrangement. Edwards and Haynes also devise a future marketing strategy for Austin and Morris ending badge engineering, again if we're lucky they hit upon the strategy that Stokes and Harry Webster came up with IOTL of Austin being a more radically styled marque with Morris as the more conventional brand
designed to fight Ford and Vauxhall. They will each have two associated brands, Austin-Healey and Vanden Plas for Austin and MG and Wolseley for Morris, each being a sporty and upmarket trim level, think of Ford's XR and Ghia brands, Riley is discontinued. A saloon version of the Maxi will be developed for sale as the Morris Major.
Edwards devises a future model strategy for BMH, IOTL all it had under development at the time of the merger was the Maxi, hence the Marina's rushed development. He also identifies an urgent need to modernise the production plants and reduce employee numbers to get BMH back into shape. News of "The Edwards Plan" is leaked in 1967 causing an outcry from the unions, Edwards approaches the Government asking for a loan to oversee the modernisation plan, say £50 million, industry Minister Tony Benn is deeply concerned at the level of job losses but is equally worried about BMH's perilous position, as IOTL he considers a merger with LMC might be the best solution.
At Leyland things have gone as per OTL except that after the takeover of Rover Stokes and Webster have bitten the bulletin and canned Triumph's V-8 telling them to adopt the Rover engine for the Triumph Stag, this will take some time for production to be expanded but the eventual V-8 Stag will be a stunning success. When he receives Benn's offer Stokes takes a much harder look at BMH and is shocked at how bad a state it's in, when he first visited Longbridge he said it looked "Half Finished!" However he sees the huge potential in the deal so he drives a much harder bargain, he insists on the Government agreeing to the loan and demands he gets full control of the business so Lyons can't protect Jaguar to the detriment of Rover and Triumph. Benn and Edwards agree to these terms, Lyons either resigns or goes along grudgingly.
Upon completion of the deal, Stokes becomes CEO and Edwards as Chairman, the Maxi/Major is looking good in testing, the car's God awful gear change has been rectified before launch. It is released in early 1969 as OTL and proves highly popular, the Morris saloon proving a better hit with more conservative British buyers while the Austin hatch sells better in Europe. Work is underway on the Marina, as OTL it will use the Minor's chassis but it will get the McPherson strut front suspension that Haynes always wanted thus preventing it being launched with dangerous understeer. The Marina wasn't actually a bad car, BL sold nearly 250,000 in one year and it was their biggest seller for most of the 1970's, its bad reputation comes from the fact that it was forced to live on long past its time because its intended replacement, the ADO77 project was cancelled in the wake of BL's bankruptcy and then it was morphed into the Ital

It's success also sees approval granted grot Haynes' Project Condor which gives birth to an MG Coupe that proves popular in America.
Haynes also modifies the Landcrab as he proposed and it too sees a sales pick up. Work now begins on the crucial replacements for the Mini, the ADO74, and for the best selling ADO16. Given the importance of the latter Stokes and Webster take a keen interest in the design, they chose Harris Mann's concept and request that it be a hatchback. With the E-Series not being the big lump of an engine as IOTL it will not distort the car's design which proves highly popular in clinics. When BL's accountants press for the Marina's heater to be fitted, Stokes, anxious to keep what seems like a potential success overrules them and orders the ADO16's heater to be carried over. The Allegro is launched in 1973 after an intensive testing programme that irons out several design faults, the Quartic wheel idea is dropped during development, it is greeted with enthusiasm by the press and becomes a big sales success, the oil shock sees a move towards more economical cars which the Allegro is well placed to exploit.
Now BL's cars will still be assembled in an atmosphere of industrial terrorism, unless Wilson hasn't shafted Barbara Castle over In Place of Strife in which case you 're looking at a radically different 1970's Britain. They will still be unreliable but even IOTL contemporary Fords, Renault's and Fiats were just as likely to leave you stranded at the roadside. The difference is that these three cars won't have got the terrible early reputation they did due to things like the Maxi's hit or miss gear change, the Marina's understeer and the Allegro's poor styling so they will have a much better public image leading to better sales and thus potentially avoiding BLMC's OTL collapse.
Feel free to blast away!