I am building a custom hub and want to know what this is.

The hub will use an odd axle that is held on by removable (replaceable) studs.

The only cruiser front wheel I have is a Worksman with a front drum, and it uses an extra-thick axle compared to normal bruiser bikes, so I know it's not the same.

The Sheldon site says 9x1mm, but I'd rather use an English size because (me being in the USA) English sizes are common hardware-store fittings that are much more easily replaceable. Problem is,,, the Harris page says the equivalent English size is 3/8-26, which matches no tap or die I've ever seen. 3/8-24 is a National Fine Standard size. Is this an error on the Harris site, or are the old English axles really threaded 26 TPI?

edit-sheldon/harris page added-

https://www.sheldonbrown.com/harris/axles.html

Since you're building the hub and using you own axle and nuts it doesn't matter what it is. Measure the existing hub that fits nicely in the fork and use the neatest common inch size, which will probably be 3/8". Standard front axles are nearer to 5/16" if 3/8" is a bit large for your fork ends, you can carefully file them open a bit, but be careful to remove metal only from the same side of the slot and not the top to preserve the location of the axle so the fork is still aligned.

BTW- if the old wheel isn't perfectly square in the fork, uou can correct it by how you open the slot if you do so.

Welcome the wounderful and confusing world of British threads; see https://en.wikipedia.org/wiki/British_Standard_Cycle. Having had various British bicycles and motorcycles when younger it makes me wax nostalgic. And yes, the Campagnolo axles really are metric millimeters in diameter and have English threads per inch.