Round and Turn are not synonyms.
From Player's Basic Rules, page 69, or the Player's Handbook, page 189 (bolding added for emphasis):
A round represents about 6 seconds in the game world. During a round, each participant in a battle takes a turn.
"On your turn" thus specifically excludes other characters' turns.
In the sections on Actions and Bonus Actions, it defines when each is allowed.
On your turn, you can move a distance up to your speed and take one action.
Various class features, spells, and other abilities let you take an additional action on your turn called a bonus action.
Then, in the section on the ready action (PBR 72, PHB 193)...
To be readied, a spell must
have a casting time of 1 action, and holding onto the
spell’s magic requires concentration (explained in
A casting time of "1 bonus action" is not the same as "1 Action". In fact, the general casting time is 1 action; a bonus action is specifically called out as different under "Casting Time" on PBR 79 and PHB 202...
A spell cast with a bonus action is especially swift. You
must use a bonus action on your turn to cast the spell,
provided that you haven’t already taken a bonus action
So, in looking at the letter of the rules, it's clear that you can't use a bonus action spell outside your turn, and further, it's explicit that your turn ends when someone else begins their turn.