If you want to know what the rules are for the tube or bus or train, its probably best you phone the Transport of London who own/run the underground. I think first bus group run the buses, but i know straight off they wont take non folding bikes, and possibly not the ones which fold in half but are still pretty big. Maybe only the brompton type size.
Most rains have somewhere you can take or leave the bike, but it will depend on how busy, and if its buy and you use it with a big bike, you will attract aggressive annoyed people.
There are many strong locks, Abus Granit, Kryptoite new york etc etc, all are pick proof, at least to the extent you'd need a lot of time, experience and be able to work unobserved, but none are cordless grinder proof.
So it kind of also depends where you lock it, and down some dark unlit street or little trafficked area then thats bound to attract thieves.
One thing is sure, you arent going to get a strong lock for £9.99 . So be prepared to invest to a certain level to get something that cant be snapped off, or quickly chewed through with a pair of cable cutters
The best thing you can do is to be insured, and to safeguard actually getting it replaced or a payout is to get a lock that is gold rated by sold secure.
Most if not all insurance companies say they want the lock you use to be gold rated by this testing company. So more than anything,more than any lock, being insured is the surefire way to go to protect your investment.