If a spell has a duration that lasts until the end of the caster’s next turn, such as booming blade, and the caster dies before then, is the spell ended when they die or does it continue until when their turn would have occurred?
The spell would last for its duration, unless it is concentration.
This applies for any spell that has a duration without a concentration, such as Continual Flame or Aid. There is no reason these spells would stop on the caster's death, so there is no reason for the spell that only lasts for 1 round to end either.
Let's take Mind Sliver for example, since Booming Blade does not last until the end of the caster's next turn, but only until the start:
Duration: 1 round
...The target must succeed on an Intelligence saving throw or take 1d6 psychic damage and subtract 1d4 from the next saving throw it makes before the end of your next turn.
As an example, suppose there are 3 creatures: the Caster, the Target, and the Ally, who have initiatives of 15, 10, and 5, respectively. The caster casts Mind Sliver, and the Target fails. The target's turn comes around and they kill the caster. On the Ally's Turn, the Ally (not knowing the caster is dead) readies an action to cast the spell Frostbite on the Target, triggering on the caster's turn provided the caster does not make use of the effects of mind sliver. When the caster's turn comes around and they take no action (because they are dead), the Ally releases the Frostbite spell, and forces the Target to make a saving throw, with a d4 subtraction due to the mind sliver* that is still active. Mechanically, this would be described as waiting until initiative 15, and if no spell is cast from the Caster on initiative 15, then the Ally will instead cast their spell.