I'm a fan of Just Mayo but I assume they do not have that in the U.K.?
My favorite go to homemade mayo is a nut based one and I loosely base my own recipe off this one:
I use way less or no oil in mine. I usually just double or triple the amount of lite coconut milk or even use other plant milks in place. If I get the ratio wrong I might add a pinch of oil to smooth it out. I honestly have made this so much I don't even measure, just throw stuff in the blender (I have a high speed blender, but a food processor works too) in estimated amounts that look right to me, and adjust as I blend. I usually make a bigger batch than the recipe and use more nuts, more liquid. I might even add a pinch of maple syrup. I have also used blanched almonds (I often cheat and buy them already blanched at the store in the baking section) in place of cashews as they are cheaper and slightly more neutral in flavor. I have used this homemade mayo to make waldorf salad and I even brought a huge batch of homemade waldorf salad and this mayo to work for a potluck and my coworkers loved it! Even a big shot doctor walked in on the potluck and tried it and was impressed, though he was hesitant at first lol.
In the U.S. I have heard of the following vegan mayo options:
Spectrum eggless Mayo
Hellman's Vegan Mayo
Nayonnaise has a low calorie version but is NASTY in my opinion. I found Spectrum eggless mayo to be very bland. Vegannaise is ok but too vinegery for me. I refuse to try Hellmans. I LOVE Hampton Creek's Just Mayo. My partner (who is not vegan) loves Just Mayo way more than traditional mayo and buys it exclusively.
I have also made avocado mayo and it is good too, but too far removed from traditional mayo for me to use it in a traditional way. I've made soy based mayo with soymilk and oil but I shudder at using that much oil and found that version too thin. I've made the tofu based mayo also but found it too bland and tofu tasting. Nuts to me are like magic for mayo, vegan "cheeses", sauces, thickening soups, and baking flaky crusts and gluten free stuff. I don't keep them around all the time though because I won't leave them alone lol. Blanched almonds tend to be safer with me as I don't tend to want to keep eating them like I do with cashews.
These things are all treats/special occasions. I keep Just Mayo on hand but don't use it a ton. My partner uses it way more.