About 10-20s, I left it on until it didn't seem to be getting much hotter. I also didn't want the battery to overheat and fail catastrophically. I think because the "wire" is such a large gauge, there's not enough current for it to get seriously hot. In a foam cutter, you're passing all that current through a much smaller cross-sectional area.

Edit: just to confirm, I did a little math. A 10cm steel wire with a tenth of the diameter would have a resistance of 5 ohms. That means that instead of 1% of the total heat dissipating in the thick wire, 80% of the heat is dissipating in the wire in foam cutter's case, and there's more total resistance, so more heat dissipation as well.

This is because:

A = π r²

R = ρ × L / A

So resistance is proportional to the material resistivity (ρ), the length (L), and the inverse square of the radius (r⁻²). That is to say, decreasing the radius by a factor of 10 increases resistance by a factor of 100.