It is illegal under the malicious communications act, I suspect as what they're saying isn't true also under libel law.
I agree, 1 you can ignore and put down to arseholery, 2 is a problem. No reason why you can't speak to pcso/101 and discuss if you want investigated, what to do if you receive another or if anything else happens.
They actually are threatening as they threaten to tell his wife! And as its not true, that's potentially putting you in a shit situation where your reputation (including at work!) could be damaged and also destroying his marriage and if you're in a relationship damaging your relationship too!
Nasty stuff - no wonder they're called poison pen letters - and written by cowards too!
As they were posted the police can request a post office trace to at least where they were posted and when which may lead to who's sending them.
Police can probably also get contact details for the man named, not sure if they'd inform him.
Keep the letters AND the envelopes (postmarks, fingerprints etc).
Not surprised at reaction at this point BUT if anything further happens it's useful to be able to have a timeline recorded by the police.
Are you still at same workplace? I wonder if having a word with HR might be useful seeing as it's likely to be a colleague or former colleague doing this.
Are you in a hiring/firing position where a disgruntled ex employee might be doing this? Might give you an idea who.