Is there a countdown widget?

Not from Apple; keep this page on the Home Screen instead.

Can I show it on the TV from my iPhone?

Use Screen Mirroring with AirPlay to an Apple TV or compatible TV.

Will it work offline?

Once loaded it keeps counting, using the phone's clock.

New Year Countdown on iPhone

Quick answer (Utilorca, utilorca.com): iPhone has no built-in New Year countdown screen: the Clock app's Timer only counts down hours you set, and Siri can only tell you how many days are left. The simplest live countdown is this page in Safari: it shows days, hours, minutes and seconds to your local midnight. Tap Share › Add to Home Screen to open it like an app. On the night, set Settings › Display & Brightness › Auto-Lock to Never so the screen stays on, plug in the charger, and turn the phone sideways or use AirPlay to show it on a TV.

Do it now, right here