When I looked the rules were pretty vague, theres a big document on the FA website
https://www.thefa.com/football-rules-governance/discipline/player-essentialsPage 9 seems to state that a suspension earned will take place immediately in every competition EXCEPT the EFL Trophy, which seems to "contain" discipline to that competition only.
ie a sending off in the EFL Trophy would lead to a ban in the next EFL Trophy game only, whereas a sending off in the League, EFL Cup, or FA Cup would lead to a ban in the next game whatever the competition EXCEPTING the EFL Trophy.