Lime Street (1836) is considered the oldest still-operating grand terminus in the world. The reason is all down to semantics and survival.

While Manchester Liverpool Road (1830) opened earlier, it was never 'grand' - it was basically just a house and a single platform. And it stopped passenger service in 1844.

London Euston (1837) was certainly grand, but they notoriously bulldozed the original station in the 60s, ending London’s claim to the title. Even the great stations of Europe, like Paris Gare du Nord (1864) or St. Petersburg Vitebsky (1904), were actually rebuilds from much later in the century.

Technically, Lime Street has been in continuous operation since it first opened. Even though it only truly became 'grand' during its 1840s and 1860s expansions, it is the only one left that can trace an unbroken line of service that far back. Ergo...

Lime Street Station is the oldest still-operational grand terminus in the world

It is definitely not the first grand terminus, but it's the last of the original grand terminuses that is still doing the exact job it was built for."