Can I buy a map at airport?

There is not a definitive answer to your question. Generally the airports and major train stations will have TIs. TIs will always have maps and sometimes better maps for sale. After that it could be hit or miss finding a shop selling maps. Personally, since maps are not very heavy, I would buy maps on Amazon or a local book store prior to leaving.

They're at the newsstands, always.

I would go with the "Boy Scout Motto" and "Be Prepared" in case they do not. Surely you can find a map of those countries before you leave. Unless, of course you are looking for a very detailed map for driving that might not be available here. Even then, you can find those "Michelan" maps at travel stores or online.

Tena, you're overthinking this one. While you are in London at you airbnb you might look on the bookshelf and find a map or atlas for London. If you don't they are at every newsagent and WH Smiths, practically on every corner. For the touristy things you will see a tourist map will do you fine in London. If you want to drop the cash and get a pocket A-Z of London it will have every street on it - 99.999% of which you will never get to. For the drive from Birmingham International Airport to Oxford any large scale map will do, easily found while in London where I said. Try WH Smith. Heck, if you have any energy before you check into the St Pancras hotel the large shopping centre in St Pancras station has both a WH Smith and Foyles bookstore. No atlas will get you around Oxford, there is just far too much detail and one way streets, etc. You will remember the previous advice of using the Park and Ride facilities. Trying to get a space on St Giles outside the pub will have you pulling your hair out. Trust me. A good UK atlas will get you close to the area. Same thing for the Cotswolds. Shame you decided against sat nav. Or sat nav app. If you decide after you are here to get one, any Halfords will sell one at a competitive price, because time is getting short now, isn't it?, and you can add North American maps to it when you get home. For driving in Scotland you need an excellent map or GPS. For people who don't like chattering GPS's (I don't either) they all have a setting to mute them and then just follow the thin blue line. That's what I do.

Here is the voice of experience. Yes, there once was a time when paper maps were the only choice. Hard to believe, isn't it, kids? Especially for big cities, with correspondingly big maps, it is important to choose one with a comprehensive street index. It's handy if the index is printed in a separate booklet rather than just down the margins of the map itself. To be clear, I now travel with a small computer and choose hotels with WiFi. Before my day's excursion, I call up Google maps in my room to find the street numbers and routes of my destination, including aerial views so I know what the places look like. No, it's not portable. It also reduces the need to stand on a windy, crowded street corner trying to hold onto a map that is wider than the span of my arms. The computer is also very helpful in sorting out routes on the Tube/Metro as well as last-minute research or even placing restaurant reservations.
Final solution: A portable, pocket-sized GPS or similar Smartphone. Sure hope the prices come down.

I'd do as Nigel said and buy at Foyle's or WH Smith at St Pancras. That would be the easiest, I'd imagine.

I would think the car rental place could give or sell you one at a reasonable price. I printed out a few prior to my trip for a few tricky places. I heavily used a pocket London AZ I bought prior online; I should have purchased theremuch cheaper. Google Streets was great for seeing tricky transitions and turns. I did paper and it was fine; my son had his I-Phone and it was bettersometimes. Whatever you use it should be what you are comfortable with before the trip.

Car places have sucko maps if they have anything at all. They're completely useless.

I'd print a route out from viamichelin.com to get you to your first night's lodging. Normally I buy maps at gas stations. When you see one, stop and pick up a good map at a good price (not sure you can get that combination at the airport). If you arrive at your lodging without having purchased a map, ask them where you can get a good road map for your trip. They'll most likely steer you to either a gas station or nearby news stand.