I never knew you could set a delay! Is this widely supported?

SRT is a text file with a awk-able format that you can just write a script to update all the timestamps of. There's also a bunch of online services that do it. VLC has it built-in (track synchronisation - you can set delays on video audio and subtitle tracks individually).

A great thing in VLC are "h" and "k" (I think) shortcuts, which shift the subtitles 100ms forward and backward.

Yes, changing the delay of subtitles is extremely widely supported and has been for many years.

At least, that's in the common Open Source video players such as vlc, mpv, mplayer, etc.