Before I begin this post, I have to point out something that a lot of Nigerians don't realise or believe. Nigerian has a postal service AND that's how most of your ASOS deliveries come (for proof that it works), so if you're for instance sending a parcel from the UK to Nigeria, and you use regular royal mail, your package will probably be fine. So, not everytime, look for someone coming back. It will take like two weeks and the girl at the post office might charge you N500 to collect it but it will probably come.
When I moved back, I made a point to look for things I can buy in Nigeria instead of spending foreign currency, because the naira is really unreliable (and that's putting it mildly). Obviously, it's frequently a lot cheaper to buy it from wherever and there are some things that are just impossible to find (matcha powder and loose leaf tea if anyone knows)
Anyway here are some sites that deliver to Nigeria;
Asos
Asos is number 1 because their delivery is FREE (no minimum spend)...as in you can buy a £3 necklace and ASOS will deliver it for free. I'm stressing this because people actually order stuff from Asos and sell the stuff, which makes me feel like people don't know delivering from asos is easy, free and relatively quick. If you're in Lagos, it's going to get to you faster sha, because they deliver to Lagos first and then dispatch to whatever state. When I ordered from Abuja, it took a while. Try and keep orders under £40 because over that and you start to have issues with customs payments and etc. Your stuff will still come but you might have to pay to collect it.
Missguided
Now, I heard about this and went to check out the delivery countries on their site and sure enough, There was Nigeria but, I've never ordered from here or known anyone that has, but if you try it, let me know what your experience is.
Lookfantastic
Look fantastic sells beauty products...look fantastic...geddit? teehee. I feel like you can find most beauty products in Nigeria, but still.
amazon
Amazon ships selects items to Nigeria and some sellers deliver to Nigeria as well.
T.M.Lewin
They have stores in Nigeria so I feel like this is moot, but if you like want the latest collection of shirts or whatever. Actually, I think ordering from the UK is actually cheaper than buying from their Nigerian stores, but don't quote me.
Aliexpress
I've actually never ordered from aliexpress. There's something about choice that cripples me, plus a sellers paradise where you cannot trust for a second that what is advertised is what you're going to get. I'll try it someday sha. *shudders*
Selfridges
Delivery on selected items. I somehow feel like Selfridges just wanted to say they deliver to Nigeria without really having to deliver on that promise.
This isn't an exhaustive list. Where do you shop?