That would make it a lot harder to determine whose property it is but it would still be considered theft to take it. The general term applied would be "stealing by finding" although it is not specifically defined in the theft act 1968 the definitions of appropriates & theft given in that act make it clear you can't take something which is not yours even if you believe the owner no longer wants it. If you took somebodys rubbish without their permission you would not necessarily be dishonest but you would be a thief.
3. “Appropriates”.— (1) Any assumption by a person of the rights of an owner amounts to an appropriation, and this includes, where he has come by the property (innocently or not) without stealing it, any later assumption of a right to it by keeping or dealing with it as owner.
Basic definition of theft.— (1) A person is guilty of theft if he dishonestly appropriates property belonging to another with the intention of permanently depriving the other of it; and “thief” and “steal” shall be construed accordingly.
(2) It is immaterial whether the appropriation is made with a view to gain, or is made for the thief’s own benefit.
There are of course exceptions, property found in a street can be handed in to a police station if it is not claimed within 28 days you may be able to claim it yourself unless it falls into one of the following categories.
The value of the property exceeds £50.00.
The property is dangerous, for example a weapon, ammunition or explosive.
The property is an official or personal document.
The property bears a clue to ownership.
You are under 18 years old
The property is suspected to be proceeds of crime
The property is a cheque, key, pawn ticket or an official or personal document.