Smart TVs all have either Freeview or Freesat tuners built into them. They would continue to receive Freeview/Freesat transmissions in the event of a loss of broadband service. (That of course assumes that they've still got power and that they're connected to an aerial or satellite dish, as appropriate).
However the 'Smart' functionality of such TVs (such as the ability to access 'on demand' TV services or view Youtube content) is delivered via the internet so, self evidently, that functionality would disappear in the absence of a broadband signal.