We know that black holes cannot form in a finite time for an outside observer. And collapsed stars will become frozen almost-black-holes. Will these objects also emit Hawking radiation, and if no, does that mean that they will exist until the end of the universe?
The main idea of Hawking radiation is that, near the event horizon of a black hole, the metric tensor can be transformed in such a way that it locally resembles Rindler Spacetime (that is, the spacetime seen for a local accelerating observer in special relativity -- see, for instance, https://centra.tecnico.ulisboa.pt/~bhw/Culetu.pdf). This allows physicists to invoke a well-known effect in quantum theory known as the Unruh Effect, which predicts that, within the framework of quantum field theory, an accelerating observer (that is, an observer in Rinder spacetime) should see the vacuum of space as being occupied by a gas whose temperature is proportional to the acceleration (see, for instance, http://www.scholarpedia.org/article/Unruh_effect).
Now, an almost-black-hole, as you call it, asymptotically becomes a black hole in time (more precisely, the metric describing it asymptotically approaches that of a Kerr black hole, assuming the star is uncharged). Thus, the spacetime near where the event horizon will eventually be asymptotically becomes Rinder space under the coordinate transformation considered above. Thus, the temperature of the gas seen by an inertial observer asymptotically becomes that of the Unruh effect. Thus, the radiation given off will asymptotically approach that predicted by the Hawking effect.
This is a pretty rudimentary analysis and doesn't actually prove or validate any of the statements given, but it's a pretty good argument based on physical intuition. I'm not certain if the radiation properties of black hole solutions formed by collapsing stars after a finite time have been studied (and even if they have, this seems like a fruitful field -- go forth and do research!), but this argument leads to a pretty good starting place for such a consideration.
I hope this helps!