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I've got a list of 313 6-digit integer-multiple anagrams, matching buzzb's calculation.

I'm attempting the rest with programmed assistance.

I meant to write "...without..."

My counts are for the anagrammed grid. For the upper right there is only 1 way to fill the initial grid, but there are 4 ways to anagram the entries so that the intersections still match.

Apologies to all, I now ger 313 too.
However my 15 was based on filling the initial grid and probably agrees with you, Buzzb

I have solved all the clues in the original grid except for one of the 4-digit entries where I have 3 possible answers,

This appears to be different from what others have managed?

Oh dear....

May I ask if the "initial grid" solves - i.e. the digits in the down and across clue boxes are the same. Put another way, does the first digit of 1 across equal the first digit of 1 down? I think the puzzle implies it does, but I can hardly believe it's that easy?

In some numericals finding the way in to the puzzle is the hardest part. That’s certainly not the case here. There is nothing misleading about the preamble. Crack away, and see if it remains that easy 😏

I've got a solution but it feels deeply unsatisfying to me as it seems like there are multiple clues where there could be multiple solutions.

Answers like 4 across, 1 down, 22 across and 17 down have 2 cells that don't cross other clues and it looks like you could swap those without violating the rules.

Also, 2 down seems to have two possible answers that, when entered in the grid, would both start with the same number (so don't interfere with 1 across) but would end in different numbers.

Does this mean I've gone wrong somewhere?

It seems like you may have misunderstood the instructions, Ipcross. Take 2d in the initial grid. It is (2:2) meaning two prime factors which differ by two in size. There are only two two-digit numbers which satisfy that, but they both begin with different digits, and only one will fit with 1a. You have to fill the initial grid first, and only then create the anagram grid from those initial answers

I have pretty much completed grid 1's clued answers, with 2 or 3 where there are more than one answer. Reading this string I'm deeply impressed that anyone has worked out that there are 313 possibilities for the long ones. I have no idea how to start to pick them apart, so a hint as to the first one or two steps would be welcome. If that's thought to be giving too much away I will supply an email address. With an orthodox puzzle it's not great disaster if one enter STRAGGLE when it should be STRUGGLE, as the error manifests itself pretty quickly ... but that discipline is wholly absent from numericals. If some kind person would be willing to verify what I have (by my giving the sum of the answers in each quadrant, for example) I'd get a chance of sleeping tonight.