From (an alleged) serving police officer I've found online. Married up to my experience of being burgled a few year ago:
[Whether a burglary is investigated] depends on a number of things, but basically comes down to whether there is any evidence or not. With most burglaries the police will look for forensics (DNA, fingerprints), CCTV, witnesses, anything stolen being sold or seen. If they have some or all of these things there is a good chance that a suspect will be identified and charged. With none of these they will almost certainly file the case unsolved on the same day.
Sometimes in the UK people will admit to previous offences when they’re caught as these can be included in their charge without dramitaclly increasing their sentence, and thus meaning they can’t be re-sentanced for another offence at a later date, and so sometimes unsolved burglaries are cleared up much later.
Fingerprints can be found that don’t identify anyone, if they have no criminal record. This is pretty rare as people don’t often start their criminal career with burglaries, or spend a lifetime of crime without being caught, but it can happen. I had a guy who was arrested for a domestic matter and his fingerprints taken and linked to a burglary that happened 10 years previously.
If there are leads police will try and solve a burglary, but alas, many, if not most, go unsolved.