I've only played one Sire bass, and that's one of the very first V7 jazz style basses that came out. C shape, 38mm nut, 7.25in radius.
The neck on that is not too dissimilar to my Squier VM Precision with regard to the thickness from back to front.
As far a I'm aware the first Sire precision style basses, the P7s, have the same necks as the V7s. There was quite a discussion on Sire's Facebook page as to what sort of neck they should put on the P7 and the prevailing opinion was stick the V7 neck on it.
Whether they're still like that I wouldn't know, and I couldn't comment on the P5 as I've never played one.
You could maybe contact Andertons, who stock Sires and Squiers and ask them about similarities and differences.