I had luck with Asda when it was down due to high traffic all day. Woke up at 2am on Wednesday and managed to get a slot for today though it was an evening one. Make sure you book a slot before filling up your basket.
Also managed to get most of the stuff on my list. Eight items were unavailable, but that's better luck than some. By 4:30am though the website started stuttering again due to people using it. I just managed to check out before it crashed.
And Wednesday evening I decided to refresh the supermarket pages I'd had up on tuesday when I started trying to do a shop (all slots fills in all shops on Tues except Asda which didn't work at all) and multiple slots came up for the very next day.
I think they occasionally manage to hire more staff and add more slots when able. Either that or there was a mass cancelling of orders.
So keep refreshing the webpages every now and again. Don't be too picky about which shop you use. Set an alarm for an ungodly hour if you get no luck during the day (especially if one of the sites is down due to high traffic as it means there's likely to be slots free once it comes back up).
And remember to book the slot as soon as you see it, and only focus on filling your basket after this. Good luck. Think of it like a game.
And if you see a slot that's too far out book it anyway and do a shop with stuff that will keep in the freezer. Then proceed to the next level of the game which is trying to get a sooner slot as well.
I know the government is saying not to stockpile, and don't go crazy with it, but just make sure you have a couple of weeks worth of food in the house to give you some breathing room in case it takes time to successfully set up your next order.