:long-winded reply only tangentially about Welsh:
I think that the defining characteristic that makes something a Jespersen’s Cycle is that a word starts off as a positive but comes to acquire a negative meaning. [EDIT - No, I’m wrong, and you’re right. But the comparison between French & Welsh on the one hand, and Dutch & English on the other still stands.] It strikes me that the Dutch change probably doesn’t count, because I take it that the niet was always negative. In fact, it looks more like English to me: Old English had a word wiht that meant ‘a thing’ (used by Tolkien in The Lord of the Rings for the Barrow-Wights), from which we get āwiht, āht ‘anything’ (MnE ‘aught’) and nānuht, nāht ‘nothing’ (MnE ‘naught’ and ‘not’ - from ‘none’ + ‘thing’). Hence:
Ic ne sǣde - I did not say
Ic ne sǣde nāht - I didn’t say nothing
I seide naught (Middle English)
I said not (Early Modern)
I did not say
I don’t know Middle or Old High German, but nicht looks so much like OE naht that I assume it’s gone through the same process.
But what’s happened with Welsh is more similar to French: in Old French a negative such as jo ne dis ‘I did not say’ can potentially be reinforced by a whole bunch of other words (which you can choose does depend a bit on the verb) - ‘a step’ (pas), ‘a crumb’ (mie), ‘a person’ (personne), ‘a thing’ (rien). Some of these are still kind of in-between in Modern French - personne is ‘no-one’, but une personne is still a person - but rien, like Welsh dim, has pretty much completed the shift from ‘something’ to ‘nothing’.
Interestingly,in this Spanish follows the same pattern as Dutch and English (and so gets trotted out to counter pedants who claim ‘I didn’t say nothing’ must mean that you did say something), whereas Catalan patterns with French:
No he dicho nada - ‘I didn’t say nothing’ (Spanish)
No he dit res - ‘I didn’t say anything’ (Catalan)
But to bring it back to Welsh – he says, aware that we’re veering madly off-topic – if you look up res in a Catalan dictionary, it has two definitions: 1. (in questions) something/anything; 2. (in negatives) nothing. And this makes sense, in that the difference between ‘Have you got anything?’ and ‘Haven’t you got anything?’ is effectively just one of emphasis, whereas the difference between ‘I’ve got something’ and ‘I haven’t got anything’ is one of fact. In a question, asking ‘Is X true?’ and asking ‘Is X true (or not)?’ and asking ‘Isn’t X true?’ are all effectively more or less the same. So res can mean ‘anything’ in a question, but ‘nothing’ in an answer, in the same way that asking Wnei di nôl panad i mi, na wnei? can mean the same as ‘Will you get me a cuppa?’