Hi.
"Red next to black" etc is just for telling two particular brightly striped American snakes -- doesn't work for anywhere else, nor even for other American ones.
Yellow on a UK snake is almost certainly a grass snake -- they usually have a yellow or white collar, with a black collar behind it. They are not striped though -- they have small black vertical markings down each side. Body colour is usually green, brownish or greyish. Eyes with round pupils, and orange or yellow irises. Up to about 4 ft, or 1.5 m. Harmless, but smells bad when upset. May hiss and strike, but with mouth closed. Eats frogs etc.
Adder is two-tone, with a V-shaped marking behind the head and a thick irregular dark zig-zag down the length of the back (not diamonds). Varies from rusty brown with reddish brown(usually female), to pale with dark brown, to whitish with black (usually male). Vertical pupils (like a cat), and up to about 18 in or 50 cm. Quite rare now. Can bite, but you generally have to pick it up first... Bite is very painful, but hardly ever dangerous for people -- can be for dogs, who are usually bitten on the face. Eats small mammals.
Slow-worm is a lizard, but looks rather like a snake. Shiny bronze colour, very smooth, with small lizard's mouth and eyelids which can close. May be striped lengthwise or uniformly coloured. Up to about 30 cm, 1 foot. Eats insects, worms & slugs.
Smooth snake is so rare you won't find it. Greyish brown with little checkerboard markings & crown mark on back of head. Found only on or near heathland. Not poisonous -- eats other reptiles, killing by constriction. About the size of an adder.
Any of these can occasionally be black or nearly black all over, but the underlying pattern will usually show through, like the black spots on a black leopard. Black grass snakes don't usually have much yellow collar.
All UK reptiles are protected by law -- the smooth snake even from capture.