What do I do to send a speedrun if I don't have a YouTube channel.

You can upload a run via google drive as well

Thanks

How long does it usually take to approve a speedrun.

Depends on how fast the run is. World Record runs for Java Edition take less than 1 hour, normally, but really slow pyramid runs take more than a week to verify.

If you don't have YouTube channel, just upload any platform (e.g. Facebook, Twitter, etc.). It depends how many days to approve your speedrun but me it just 5 days.