5mm away from pipe is perfect, where one is above the other put cable below pipe.
You cannot chase it into wall except in a safe zone. In your case from information provided that is within 150mm of the side wall adjacent to the wall you want to run your chase inside.
Well done for using 10mm and for asking us.
Running it inside a warm area is not ideal, chasing it on the other side of the wall within 150mm of the corner would be better from the point of view of heat loss. If you do decide to run it inside the hot area, clip it to wood the cable sheds heat into the wood quite effectively. Whatever you do don't let it go for any part of it's journey surrounded by insulation. If it has to traverse a loft, run a plank above the insulation and clip it to the plank.
To actually do the work itself in the room with the shower you do have to be qualified aswell as a member of a certifying body. You can't do the work and get it checked, you have to hand the job over.
I would have no objection to you first fixing the cable as long as I could inspect it throughout it's length and you run it as I design, but I wouldn't be able to sign off your work in the shower room, I would have to actually do that work to comply with approved document P..
Don't shoot me, I didn't make the rules, but as a full scope member of a certifying body I must observe them or it makes a nonsense of the £3,000 of training equipment and membership fees I have had to endure in the interest of your wellbeing.
By all means ask on the electrical forum but you will be dealt with swiftly and pugnatiously.
Bear in mind that should bathroom bonding be appropriate in your case it has to be caged to the shower.