If you are indeed in the UK then I think it will work fine. Make it good and predator proof, which means ensuring no digging under the run, perhaps with a skirt about 18" wide of hardware cloth extending out into the grass (you'll be able to mow over it soon enough.) In that climate you don't ever need to lock them in the coop - this is almost never necessary just for weather, except in extreme climates. If I were going to build this I would make the run predator proof and never close the coop.
Yes, light is only to increase egg production for the short day part of the year. Not at all a necessity. Again, in the UK (and nearly all of the US) you do not need to provide heat for grown birds. Really.
One of the most difficult parts of building a small, short coop is providing enough ventilation without creating a draft on the roosting birds. Basically it needs to be a hole in the top of the roof which is weatherproof -- you see what I mean. If you build it with a slanted roof and place things right, the vent can be under the roof, on the wall, on the high side. Probably needs to be the width of the wall and maybe 4" tall -- I hesitate to type that because every coop is different.
Perches and other things to jump on or climb up are great in runs. You might find that a sand run would work best for ease of cleaning and odor control. Also gives them a great place to dust bathe.
My biggest concern is not 4 different breeds but not raising them together. If power would not be too difficult, perhaps the solution is to reconsider raising chicks and do so in the coop. Even if you bought 4 older birds of the same breed, but from more than one flock, you are going to have integration issues, and these can get ugly fast. Usual recommendations for adding new birds include things like housing them separately but next to each other for a matter of weeks. If I were in your position I would limit myself to either buying day olds or buying older birds all from the same flock.
Perhaps you could buy day olds and have someone raise them for you. You could undoubtedly knock together a breeder box in part of a day, probably from scraps, if needed. Not too unusual for people to do this for a fee.
Best of luck!