With the strict rules about liquids on planes you could assume that reusable drinks containers are not welcome either. This turns out not to be the case. Your waste reduction ethics do not have to be left at the terminal door.
Travel forums abound with people who take empty bottles through all manner of airports with no problems. The most that could happen is security staff wanting to double check that it’s empty. To be sure that security staff know that your water bottle or coffee cup is empty just take the lid off before you place it in the tray for scanning. Some people in forums do seem confused about the partially empty rule i.e. if you have a 200ml tub of toothpaste, but it is about half empty, you will not be allowed the tube. This is because the security staff do not have time to accurately assess the exact amount of toothpaste in your tube. However, if your water bottle or keep cup is completely empty it is no different to the bag they give you to put your toiletries in at the gate, a vessel that can contain more then 100ml of liquid but currently doesn’t.
With many airport lounge franchises only providing paper cups, and the airlines mainly giving plastic and paper cups in economy, you will get lots of use from your cup and bottle. Just remember to carry a washable napkin with you to soak up any dregs if your vessel is not spill proof, as you may not get to a bathroom to give it a rinse before you need to out it back in your bag!
Many airports have water fountains after security, or a number of the cafes will allow you to fill up your bottle. Check out this great listing of where the water fountains are located at all the major UK airports from Money Saving Expert.