If an intro cut-scene has already been shown (to set the theme), how do you then add the main character's backstory without taking up more of the player's time before the action start?

So you have determined that there is a time limit for a cut-scene and/or backstory before starting game action. If the protagonist's backstory truly is vitally important right from the get-go, then replace the thematic cut scene with the protagonist's backstory. Or make it part of the cut-scene. But it's common to reveal the backstory bit by bit as the overall storyline (in this game, delivered through gameplay or cut-scenes) takes shape.

Another option is for the cut-scene to take place during the boot/title sequence, then show the vitally important backstory after the player presses Start.

Thread has been moved to the Writing forum, by the way.

Thanks for replying, Tom.

I've been reading books on game design by industry experts, and they say you shouldn't drag out the intro for longer than (usually) one minute, as players get impatient and tend to skip long intros, The intro cut-scene is vital in hooking the player into the setup of what's happened and what they should do next, but this will take at least 30 seconds. It might take another 60 seconds to show the protagonist's back story. I know 90 seconds doesn't sound like a long time but in game time, especially if you're waiting to jump into the action, it could feel like decades.
Even if I show the thematic scene at the very start, an extra cut-scene straight after that would just delay the action and could even feel ‘stuck on’. Hence finding your site in the hope of some professional game dev advice regarding this ?
I think I've worked out how to do it - maybe this will help others with the same issue.
The first objective is to get to a building being used as a base, When the player gets there, he / she's questioned on their motives (why are they there? what do they want? etc.)
This will let us show the protagonist's back story as the questions are asked, even if it takes 60 seconds or more. The player will have been attacked on the way to the base, so the questions will give them some calm time before they set out into the next, probably dangerous scenario.