...describe uncountable nouns as being singular (this is irrelevant as uncountable nouns are always singular).
Well, of course - this is the accepted grammatical rule and you'll find it in all the grammar books: Nouns can be countable or uncountable. If countable they express the concept of a discrete object or entity, of which there can be more than one (ie we can "count" them - one cat, two cats. three cats...) and the word can be used in both the singular and plural (cat/cats), which will be evident not only from the use of the plural -s suffix, but also by the concurrent use of singular or plural determiners, pronouns, subject verb agreement etc. Eg.
That cat looks like John's cat. It's lovely / Those cats have already eaten / There aren't many cats around here
On the other hand, uncountable nouns are always singular and describe not a discrete object but a "mass" - eg oxygen, information, furniture - or an abstract quality - advice, anger, knowledge, news. These don't have a plural form and are consequently always used with singular determiners and verbs. Eg:
That is fake news but not *Those are fake newses
There isn't much oxygen left but not *There aren't many oxygens left
... and so on. Nothing new here. I've discussed elsewhere (see here) whether it's valid to say that nouns are countable or uncountable, or whether we shouldn't rather think of them as being used countably or uncountably. But that's not the issue here. What I want to look at is whether, as Cambridge assert, that an uncountable noun can never be plural.
Think about the following nouns(1) (and there are many others that could have been added to the list):
goods remains leftovers odds throes thanks groceries jitters hysterics clothes surroundings entrails amends valuables arrears scissors shears tweezers bellows pliers tights jeans trousers...
There's no question that these words are plural. Apart from the -s suffix, they meet our criterion of being used with plural determiners, pronouns and verbs...
Those scissors need sharpening / There aren't many clothes to throw away / The surroundings were beautiful / His arrears have decreased over the past three months. He should clear them soon
... and they can't be used in the singular. You can't say : *She had a hysteric / * That good you ordered online has arrived / *I sent him another thank for his help.
So - grammatically they're always plural. But does that mean they are also countable? If they can't be used in the singular, then obviously you can't talk about *one remain etc. Even with the words like jeans and scissors (which Quirk et al (1972) categorise as summative nouns) the fact that the object is composed of two identical parts doesn't mean that it can be split into two. If I take the screw out of the middle of the scissors and separate the two parts, I can't wave them in the air and say *Here I have one scissor, and here is another scissor.
But the words can't be used with plural numbers either. You can't say *She had two hysterics during the morning / *I have three scissors. You need to add a word which "itemises" them : She had two fits of hysterics during the morning / I have three pairs of scissors But the countable words are fits and pairs - not hysterics or scissors - and this option isn't possible for most of the words on the list above.
So for me, and despite Cambridge's assertion, these words are plural but uncountable. Literally, you can't count them. In fact, Quirk et al avoid the terms countable/uncountable altogether - both for singular and plural nouns - and instead talk about variable and invariable nouns(2), thus resolving the problem.
Notes and References
1. Obviously, what I'm saying refers to the words with certain meanings - for example, groceries. The discussion holds when the meaning is items of shopping but not when the word refers to the shop where the goods are bought - eg She owns a grocery in Plymouth. Be careful too that you don't confuse the word class. For example, the word leftover exists - but as an adjective: I'm sick of eating leftover rice every day. Say the same thing without specifying the food item, so that the word becomes the noun, and it must be plural: I'm sick of eating leftovers every day.
2. Quirk et al, in A Grammar of Contemporary English (Longman 1972), devote six pages to the area of "invariable" nouns, of which four are devoted to invariable plurals. They also split them into a number of categories. For example, under singular invariables, as well as the different types of "mass" nouns, they also include a list of games ending in -s: billiards, bowls, darts etc. They also deal thoroughly with "exceptions" like the one mentioned for groceries above.
