It seems as though the Timurids are not well known at all. The original heir to the throne of Timur was supposed to inherit the entire empire without a fight but then he died, and the next two heirs died so eventually the grandson was chosen. So basically no pretenders and relative peace and that would mean a longer lasting nation than in our timeline. Perhap Timur's descendants industrialize and eventually make it a gunpowder empire. And after centuries of rule it eventually collapses as the Manchu, Persians, and possibly Russians and Ottomans sieze the land. Timur will forever go down as the reincarnation of Genghis Khan.
I am afraid that at any given moment the Empire of Tamerlane was destined to be in some kind of turmoil after his death.
No matter when he dies there must be some squabble in the Empire because that's the way he created it. All the military power belonged to Tamerlane; he gave provinces to his sons and grandsons but (and that is a very important 'but') their military commanders were responsible
directly to Tamerlane himself, which made his sons/grandsons mostly figureheads (at least military).
That meant that Timur's (grand)sons cannot go to their soldiers and give them any command, no, they gave their order to their commander-in-chief and if he thought it appropriate he gave the command to the troops. But if the (grand)sons' commander-in-chief thought that this command is not consistent with his instructions from Tamerlane - than no soldiers did anything.
That system was perfect to prevent any attempt of usurpation/revolt/disobedience from Timur's (grand)sons. But when he died he left his successors without any real experience of independent military command with a bunch of seasoned famous warlords who owed personal loyalties of their soldiers. Which sometimes looked like a puppy trying to lead a pack of wild wolves.
The tradition of Chagatai Ulus (which was an ethnic core of the Empire) was an old Mongol custom of a free will of a free Mongol to switch loyalties whenever it suited him, there was no dishonor in it.
Of course the warriors worshipped Timur and part of his fame his (grand)sons inherited, but there was one more problem - Tamerlane's rule did not have any steppe legitimacy, he was not Chengizzid and consequently neither he nor his successors had any right for a title of Khan. Which did not make it any easier for his bloodline to keep power as they were just bloodline of a warlord, very famous and charismatic general, but still... It was not a gold bloodline of Chengizz Khan, the only royal bloodline that mattered in steppe.