Ask for centimetres, not the size name
Before you order a mattress for a frame you already own, write down the frame's inside width and length. Then ask the seller for the mattress in centimetres. A UK king (150 × 200 cm on the Which? table) is close to a US queen on the seller charts (about 152 × 203 cm) and not close to a US king (about 193 × 203 cm on those same charts).
An Australian king on those seller charts is about 183 × 203 cm. That is wider than a UK super king, which Which? lists at 180 × 200 cm, and narrower than the US king figure above. Close is not the same. A fitted sheet from the other country may not stay on.
This page does not rank a brand that sells any of those sizes.
Sources: Which?, KAS Australia, Ecosa.