From everything I've seen online, it seems not, but I've heard a few credible individuals claim that they are allowed to do that. I think there must be some kernel of truth in it, or I wouldn't have heard multiple credible people say that.
If it's never legally permitted for a landlord to do this in the UK, was it ever? When did it stop being the case? If it is allowed, what circumstances is it allowed under?
Ultimately yes, this can and does happen, but there are a few steps necessary before the bailiffs come to your door.
In times gone by, the common law recognized a right to "distrain for rent", meaning that a landlord could come round to the tenant and seize some property as security (up to the value of the rent owed). If the rent is not then paid within a certain time, the goods could be sold. Various additional provisions of law covered the circumstances around forcing entry, breaking open locked cupboards, and searching off the premises for goods which the tenant had concealed. There were also special-case rules about what could be seized, such as a statute of the seventeenth century saying (among other things) that landlords could not harvest growing crops, but could take already-cut sheaves.
Subsequent reforms for the protection of tenants have made it so that landlords cannot do this at their whim. They first have to go to the County Court for a judgement that there is an unpaid debt. (In fact, this and subsequent steps can happen with any debt.) On that basis, they can apply for a "warrant of control", under which an "enforcement officer", commonly called a bailiff, will take charge of collecting the money owed. There are quite a few steps and rules here, but the basic position is the same in terms of being able to take your stuff. And it is not the landlord walking around making those decisions, but a court-appointed officer.
The previous common-law "distress" process is now totally abolished, for most tenants by the Housing Act 1988, and for everybody in 2014, because of the Tribunals, Courts and Enforcement Act 2007 section 71, which simply states
The common law right to distrain for arrears of rent is abolished.
Commercial property has its own legal regime since that Act, but previously functioned in the same way.
So landlords can no longer seize your property on their own, but they can go through a few hoops and ask a court to do it. They also do not get to keep the specific property, but just get the money. As mentioned above, this is the general avenue for what happens when a court orders you to pay money, and you don't do it. The High Court has its own enforcement officers who are able to deal with larger amounts of money, as well as enforcing evictions.
In Scotland, the same basic pattern applies but different words are used, and some of the detailed rules are different. "Attachment" is when a sheriff officer (= a bailiff) comes round to value your possessions and auction them off if you don't pay. There is also "arrestment" which is taking money out of your bank account.
