When I want to return to London, will Spanish authorities think that I've overstayed if I show my UK passport?
Yes! This is a great reason not to present your UK passport. You can also present both, which seems like it would definitely be useful after the EES goes into operation; it would give the officer the chance to delete the records associated with your UK passport from the system. As of now, however, the EES is not in operation, so the primary record of your entry is the stamp in your passport, and the officer is likely to refuse to give you an exit stamp. Still, it may be worthwhile asking for one.
(Note that stamps showing a past stay of over 90 days shouldn't be a problem; for example, you might have had a residence permit during that time.)
In essence, will I then be required to always using my Bulgarian ID whenever I want to fly from the UK to Spain?
Not necessarily; you can still use your UK passport, though there's always a chance they'll want to go after you for the overstay, at which point you will want to mention your dual nationality to get them to drop it.
The worst likely outcome for traveling without your Bulgarian ID is probably a provisional fine that you would be able to avoid having to pay by sending in evidence of your Bulgarian nationality. In many cases, however, you might be more likely to be able to talk your way across the border -- especially on departure.
Also consider that carrying and using your EU passport or ID card isn't only a matter of convenience for yourself; it's also courteous to the immigration inspectors, who have less work to do when they process an EU citizen than they do for third-country nationals.
In short, life will be simpler if you maintain a valid Bulgarian ID or passport and use it when you travel to the EU or Schengen area. If, for some reason, you find yourself wanting to travel with only your UK passport, it should be possible even if there is a greater risk of complication.