If it's a tenement, unless your title deeds say otherwise, then the Law of Tenement applies and stairs and landings are common and all properties (excepting any main doors without their own access to the stair) are responsible for the upkeep. They are not private places for any one individual resident i.e. "your" bit of landing outside your flat is actually everyone's bit of landing.
Again this is all conditional on your title deeds not saying otherwise.
It is very unlikely that the title deeds confer any right of storage in the stairwells except in any specific storage areas, e.g. cupboards under the stair.
If anyone is dumping / abandoning in your stairwell then the first approach is to confront the person responsible and politely remind them of their responsibility to keep the common areas clean and free of obstructions. The fire brigade are very clear on this - bikes are a hazard in common stairs, both for residents and any personnel responding to a call (stairs are likely to be smoky, they're wearing bulky gear and they don't need hazards to be falling over or getting their equipment snagged in).
Beyond that I would take a progressive approach. Get a reasonable number of neighbours on board and decide what level of objects you can tolerate in your stair and where. Each tenement stair is different, some can accommodate many bikes without problems, others a few, some none. Put up some signs clearly explaining where bikes can and can not be stored. Make it clear that any bikes acting as an obstruction will be reported for removal. Go from there.
Clearly abandoned bikes I've taken an iterative approach to before. Marked them up. Waited. Marked again. Waited. Then the sign goes up saying "unclaimed bikes will be moved". Then I've but the lock, but left it hanging there. Waited a bit more. Then the bikes went in the back garden. Then waited a bit more as the rain finished the job and turned them to rust. Then they went further and further up the garden before they were consigned to a skip. One new neighbour did object to a very inconsiderately left bike being moved (it wasn't locked) and put a note up saying that nobody had any right to "interfere with their property". They got a barrage of notes in response from other neighbours reminding them of their rights (i.e. none) and next time the bike was going in the back garden and if they persisted then their landlord would be contacted.
Regards to a main door flat dumping in the stair, I'd have no problem telling the owner that if they don't move it you'll "have it removed" for them as it's an obstruction and they have no legal right of access or storage in the common stair.
If you cut a lock off a bike that was obstructing a stair, after making a good effort to identify the owner, I'm not sure a/ what crime you could be committing given you are cleaning/maintaining the common area as the law states you have a duty to do and b/ I'm not sure if anyone complained the Police would care to follow it up. You aren't stealing the bike, just moving it.