So what if it's "traditional"? Your wedding, your rules! No speeches from either parent.
I'm guessing from the sheer number of people who are trying to walk all over you that you are in general a bit of a People Pleaser. This is no criticism of you OP; adults who have ended up as People Pleasers have generally had the sort of childhood where the adults around them have made their life very difficult, and as children they have tried please those adults to keep them from being aggressive / sulking / downright nasty.
Unfortunately some people take your trying to please everyone around you as permission to push you around - that's why you should be aware you have this tendency, and practice saying NO to them. And whenever they ask for something (or, I suspect, try to paint you into a corner to force you to agree to their unreasonable demands), your default answer should be NO before you even consider what they're even asking of you.
So, the wedding and the people trying to push you around -
I would consider not having either your mum or your dad there. You would "like an opportunity to let our hair down and have a fun party"? Disinvite them. Seriously. They sound like a pair of total wankers and instead of being able to let your hair down you will be on tenterhooks waiting for one or both to embarrass the hell out of you; so I would consider heading that off at the pass by telling them that they've given you so much trouble in the run-up to the wedding that they are disinvited. That takes care of your dad's wife too, she can sit and drink herself insensible at home.
Of course you have your grandfather walk you down the aisle. And if you really must have a speech, he could do that too.
Tel your SIL she's being ridiculous. She can come as a guest or not at all.
And of course, tough shit if someone's kids love a party - they can throw a party for them themselves. This is YOUR party and you control the guest list and they are NOT invited. If that's a problem for the parents, you will absolutely understand if that means they can't come either.
Time to practice your deathstare. The one that means you don't have to verbalise 'fuck off', your face does all the work for you.