I left mine until about three weeks before, which I don't recommend. What finally got me moving wasn't a template - it was writing down three specific memories before trying to make any of it sound polished.
Once I had those, the shape was obvious: who I am to him, one story, one honest line about the two of them, then the toast. Four or five minutes at most. The generic stuff ("he's a top bloke and a great mate") is true of everyone, so it just slides off people.
For the very first draft I used a wedding speech generator purely to get something on the page - I rewrote nearly all of it, but arguing with a rough draft is much easier than staring at a blank one. Then I read it aloud about ten times, which is where the real editing happened.
Does anyone else find the jokes only work once you've said them out loud a few times?