High temperature superconducting (HTS) transformers windings operate subcooled at atmospheric pressure to temperatures as low as 65 K, with minimal power dissipation from AC loss in normal operation. During short circuits, HTS transformers are subjected to transient heating by currents as much as 10x rated current for durations of up to 2 s.After isolating the fault, HTS transformers are required to cool back down to their base temperature while carrying the operating current. HTS transformer conductors are insulated, typically with wrapped self-adhesive polyimide tape, and windings consist of close packed conductors wound on composite formers.In this paper, the heat transfer, transient thermal response, and recovery performance of brass-laminated coated conductor HTS wire as well as metallic conductors in liquid nitrogen (LN2) were measured at a range of temperatures from 77 K to 64 K and pressures from atmospheric to 0.14 bar in heating and cooling situations. Measurements were made on bare tapes, and on tapes wrapped with polyimide insulation tape and with aramid paper as well as polymer coated tapes with a range of coating thickness. Heat transfer from tapes mounted with one face in contact with a glass-epoxy composite cylindrical former was measured for comparison with free-standing tapes. Using conductors with solid polymer coatings of optimized thickness immersed in subcooled LN2 results in the highest heat transfer and fast recovery following heating to 300 K by a high current pulse. Compared to a bare tape in LN2 at ambient pressure at 77 K, the heat transfer in a coated tape in 65 K subcooled LN2 can be very significantly enhanced, by up to a factor of 15x, and recovery can be 7x faster.