In short,
- it is an offence to steal a hat (no surprises there),
- it is an offence to possess a hat when you know it was stolen, for example because you stole it yourself,
- if you are in certain regulated jobs, such as in the financial sector, and you learn or suspect as a result of that job that someone has property they obtained as a result of crime, you have to report them.
Therefore, it's an offence in this situation not to report the stolen hat. Moreover, that whole set of rules is "about" money laundering, and so the possession of criminal property is a money laundering offence, even though the person might not be trying to launder anything. However, stealing a hat is not the worst crime ever and it is unlikely that the anti-money-laundering regime would be brought to bear on a simple shoplifter. It is more salient for people in the regulated sector who are being asked to look out for all sorts of criminal-adjacent conduct.
Section 329 of the Act does say that it's an offence to have knowing possession of criminal property. That includes goods which a person obtained as a result of their own criminal conduct, such as theft. The hat is such property even if its value is small.
Section 340(11)(a) of the Act says that
[Money laundering is an act which] constitutes an offence under section 327, 328 or 329
and so the offence in section 329 can indeed be classed as "money laundering" for the purposes of Part 7 of the Act. The effect of that classification is to engage the failure-to-disclose offences in sections 330 onwards, in relation to somebody knowing about the conduct. This interpretative provision does not endow the words "money laundering" with this meaning, outside the context of the Act. The net effect is something like: if you are somebody who has to watch out for possible money laundering, then one of the things that's reportable is suspecting someone of being in possession of stolen goods.
So, for example, I might be a casino operator. I am "in the regulated sector" by virtue of Schedule 9, s.1(1)(r). In the course of business I come to suspect that my customer is in possession of stolen property, and I don't report it. Then I may be committing the section 330 offence of failing to disclose money laundering. While the statute is presumably intended to cover situations like someone coming to my cash desk with stolen money, swapping it for chips, and later reclaiming different physical notes, the offence as written is more broad. Indeed, on its face it does take in a very wide swathe of scenarios where [somebody] has [some stuff] as a result of [a crime], even someone walking into my premises with a hat that I recognize as stolen.
Where all this becomes a bit more reasonable is in the wider context of prosecutorial charging procedure. The linked CPS document refers to some of the relevant considerations. For example, it notes:
The Court of Appeal has considered criticisms of the use of (in particular) section 329 of POCA rather than an offence of handling stolen goods in some cases. This is particularly the case where the defendant is found in possession of only a small amount or value of stolen goods that do not immediately suggest that the defendant is routinely involved in money laundering activities. [...] Prosecutors should identify the most appropriate charge to reflect the suspect's criminality, rather than consider the provisions of sections 327 to 329 of POCA as providing an alternative to offences of handling stolen goods in all cases.
The idea is that the criminal charges should reflect the nature and gravity of the alleged conduct. A shoplifter could be charged with theft as that is a more natural fit for what happened. The offence of handling stolen goods (or in Scotland, "reset") is also apt for a situation where a prosecutor might not be able to prove that the person actually did the theft, but can show that they accepted the goods in the knowledge that they'd been stolen.
In the same way, section 327(1)(a) of the Act makes it an offence to "conceal criminal property", but that doesn't mean it's an appropriate charge for a burglar putting things into a big bag marked "swag". That particular conduct is ancillary to the burglary itself, which would be the main offence actually charged.