Can you activate a planeswalker loyalty ability the turn it enters if it wasn't cast?

Planeswalkers do not have summoning sickness in the traditional sense — that rule applies only to attacking and to activated abilities with the tap or untap symbol (CR 302.6, CR 303.4). Loyalty abilities use a completely different timing restriction.

Under CR 606.3, a player may activate a loyalty ability of a planeswalker they control only once per turn, and only during their own main phase when the stack is empty. Critically, there is no requirement that the planeswalker was cast or that it has been under your control since the beginning of your turn.

So if a planeswalker enters the battlefield through any means — a spell, an effect like Oath of Nissa, a reanimation effect, a 'put onto the battlefield' instruction — you may immediately activate one of its loyalty abilities during your main phase on the same turn, provided the stack is empty and you haven't already activated a loyalty ability of that planeswalker this turn.

Example: It's your main phase and the stack is empty. You use Atraxa, Praetors' Voice's proliferate trigger from a previous effect to put a planeswalker onto the battlefield, or simply your opponent's effect puts Liliana of the Veil under your control. You can immediately activate one of Liliana's loyalty abilities that same main phase — no waiting required.

The only restriction is the once-per-turn limit per planeswalker (CR 606.3) and the main phase / empty stack requirement. How the planeswalker arrived is irrelevant.