This is an interesting question -- The simplest answer is that the earth would continue in its elliptical orbit despite the missing sun (which the Earth wouldn't know about, as you say, for 8.5 minutes).

While this might seem crazy, consider the inverse problem -- image a sun 'appears' at some point in space -- how long before objects are attracted to it?

A possible scenario is a star orbiting a binary star whose components orbit each other rapidly -- (say a pair of neutron stars or, better yet a pulsar). The orbit of the far component should have a phase lag -- i.e. there should be a difference between a Newtonian orbit and an Einsteinian orbit, partially due to the time lag between the star positions and positions as seen by the outer star. Some of these effects have been observed -- there is a famous pulsar, neutron star pair which has been observed for decades, just to validate general relativity.

See:
binary pulsar
astrosun2