Here are some Welsh cakes that I made yesterday. They are very easy and delicious with butter. :)
Here is the recipe:
4 oz self-raising flour 2 oz butter or margarine 1 oz sugar Pinch of mixed spice 1½ oz dried fruit of some kind (currants/sultanas/raisins) A drop or two of milk to mix
Rub the butter into the flour. Add the sugar, spice and fruit and mix them in. Add a drop of milk to bind the mixture together into a soft dough. Roll out the dough until it is about half and inch (1 cm) thick. Cut into circles with a pastry cutter. Cook on a griddle or heavy frying pan until done, flipping occasionally to ensure even cooking.
You need the pan or griddle as oil free a possible, but obviously you don't want them to stick. What I do is to apply a drop of oil to the hot griddle then rub it around with a piece of paper kitchen towel, wiping it all off in the process and just leaving the merest trace of oil on the surface.
"Mixed spice" (according to the jar) is a mixture of ground cinnamon (cassia) 40%, ground coriander 38%, caraway, nutmeg 4%, ginger and cloves. It's a ready mixed spicy flavouring that is commonly used in cakes, puddings and biscuits in the UK.
They are often eaten as part of a Welsh afternoon tea. Because they are so quick and easy to make and you cooked them on a griddle rather than in an oven, they can be made fresh. The mass produced ones sold in shops are never as nice.
Margarine is absolutely fine. In fact I generally use it rather than butter. Water would probably be OK, though I've never tried it. You only need a very little, like when mixing pastry.
Otherwise I don't see why you couldn't use the milk substitute he generally drinks, soya, oat milk, whatever. Alternatively, you could try a little beaten egg. One of my Welsh cake recipes uses egg and milk for mixing.
I did wonder whether eggs might also be a problem, but thought I'd mention that option anyway. I'm sure the oat milk will be absolutely fine as a substitute. It only takes a splash of liquid, just enough to moisten and bind the mix together so it shouldn't alter the flavour.
It's not exactly fried, it's baked on the griddle one side at a time. Only the tiniest film of oil is required, just enough to stop them sticking, like you would grease a try when baking scones in the oven. I moisten a paper kitchen towel with a drop of sunflower oil and rub that over the griddle. You could use butter, but it might burn at the temperature required for baking.
Basically, it's the ancient "cooking things on a hot stone over an open fire" method that has been used for thousands of years for making flat bread and thin cakes of all kinds. You can in fact do ordinary scones the same way, though you need to roll them out thinner in order to make sure they cook right through.