Theft requires intention, so whether Bob has committed theft will turn on the details. Honest belief that property has been abandoned is a defence against theft.
Your case is superficially close to Williams v Phillips (1957) 41 Cr App R 5
where rubbish was left for collection by the local authority refuse workers. The Divisional Court held that such refuse remained the householder's property until collected. Hence, refuse workers helping themselves to such property are guilty of theft, on the basis that the property never became ownerless. From the judgment:
The first point that is taken here, that the property was abandoned, is on the face of it untenable. Of course, that is not so. If I put refuse in my dustbin outside my house, I am not abandoning it in the sense that I am leaving it for anybody to take away. I am putting it out so that it may be collected and taken away by the local authority, and until it has been taken away by the local authority it is my property. It is my property and I can take it back and prevent anybody else from taking it away. It is simply put there for the Corporation [the employer of the dustmen] or the local authority, as the case may be, to come and clear it away. Once the Corporation come and clear it away, it seems to me that because I intended it to pass from myself to them, it becomes their property. Therefore, there is no ground for saying that this is abandoned property. As long as the property remains on the owner’s premises, it cannot be abandoned property. It is a wholly untenable proposition to say that refuse which a house holder puts out to taken away is abandoned. Very likely he does not want it himself and that is why he puts it in the dustbin. He puts it in the dustbin, not so that anybody can come along and take it, but so that the Corporation can come along and take it.
However, it must be noted that that case involved workers taking rubbish they were tasked with collecting, not with a passerby helping themselves before the refuse was collected. My read of this case is that the remarks above were coloured by this context, and would not vitiate a defence of abandonment in your case, as later case law demonstrates.
Honest belief that property has been abandoned is a defence against theft. In R v (Adrian) Small [1987] Crim LR 778, a conviction of theft of a car was quashed, as the accused believed the owner of a car that had been left for over a week with the keys in the ignition had abandoned the car. Importantly, the Court of Appeal found the requisite belief is subjective and does not need to be objectively reasonable. In your case, Bob could be accused of theft but a defence of abandonment is likely to succeed unless other facts could be proved, such as:
Bob was an employee or former employee of Abdul's or the publisher and knew of the arrangement with the publisher.
The newspapers were presented on the pavement in such a way that would make a claim of honest belief of abandonment improbable. For example, if the papers were tied up with string, or had a note saying 'for collection' secured on top.(If the papers were placed with just a rock on top, abandonment would be likely to be an honest belief, as Bob may think the rock was there to avoid the abandoned property becoming litter, though interestingly one could argue that taking the rock would be theft, since it's serving a purpose as a paperweight.)
Bob had been asked to return the paper by either Abdul or the publisher after taking them.
Even if it was widely known among newspaper agents that papers left outside after the day's trading were part of a commercial arrangement with the publisher, as long as this was not known to Bob, the defence succeeds.
In my view, the judgment of Williams vs Phillips has a too narrow definition of abandonment. Rubbish left out for collection in the ordinary course of events is abandoned property. The fact that the owner expects the refuse to be collected by an entity is not the same as intending to transfer ownership to that entity, nor as retaining ownership until that point. If the owner is indifferent about how the rubbish is disposed of, then the rubbish is abandoned.
US and Canadian law seems to endorse this view. As Saw Cheng Lim in "The law of abandonment and the passing of property in trash" notes:
From a brief survey of US and Canadian case law, it is apparent
that two requirements must be satisfied in order to effect a proper
abandonment of property. … There must therefore be, in addition to the overt act of abandonment itself, a specific intention/motive on the part of the
original owner to completely relinquish all rights of ownership –
voluntarily and, more importantly, without regard as to who may
subsequently take possession of the property. It bears repeating that
such relinquishment must be to the extent where the former owner is
completely indifferent as to the fate of the discarded object (ie, as to
what/who may await the abandoned property).
Thus, if rubbish is left out with the expectation that it will be collected, but the owner is actually indifferent about who collects it, then the rubbish is abandoned, contra Williams. Furthermore, R v (Adrian) Small suggests that even if the owner is not so indifferent because of an arrangement he has with the publisher, a passerby of the evening who sees that morning's papers left on the pavement would be entitled to claim abandonment and it would be up to the prosecution to prove that that belief was not honestly held. If Bob "rescues from vain destruction" the paper, he would not be guilty of theft.
See also Can There Be Theft Of A Discarded Item (2021) by Benny Tan Zhi Peng, regarding Parti Liyani v Public Prosecutor [2020] SGHC 187.