First off, as /u/DisheveledLibrarian said, the "last" Roman emperor, Romulus Augustulus, was not executed. Although his father Orestes was, and so was the Romulus' deposer, Odoacer.

There is an article by Michele Salzman that answers this question as well as can be answered given the limited information of the late 5th century. In summation: Their life changed not at all, and in some ways, for the better. Basically, after the collapse of a stable imperial dynasty in the west after the assassination of Valentinian III, Italian/Roman senators actually increased their influence and power, especially as these weaker emperors began to actively return to Rome in order to build a consensus to rule with the only other players left with any political clout outside of the army, i.e. the aristocracy. In fact, it was supposedly a delegation of senators that was sent to Constantinople to ask the emperor Zeno to accept Odoacer as the new military leader of the west.

This senatorial power in fact only increased after Theoderic disposed of Odoacer. Theoderic actively courted the Roman Senate, and tried to build an image of himself as a collegial princeps, a "first among equals" with the aristocracy, and to portray himself as effectively a western emperor of old. In fact, a very good argument can be made that the person who really destroyed the Roman Senate, was Justinian, as his invasion of Italy and subsequent Gothic War got all those powerful Italian senators killed (of course the other argument is it was the Goths, since they were the ones who killed those senators).

So to re-answer your question as to what life was like after the last emperor is deposed in 476?

• The next day, your day would be exactly the same
• A year later, it would be better.
• 2 years later, even better.
• 40 years later, THE BEST.
• 80 years later? We're dead. We're all dead. Italy is dead. Everybody and everything is dead.

Sources:

Arnold, Jonathan J. 2014. Theoderic and the Roman Imperial Restoration. Cambridge: Cambridge University Press.

Gillett, Andrew. 2001. “Rome, Ravenna and the Last Western Emperors.” Papers of the British School at Rome 69: 131–67.

Salzman, Michele. 2017. "Emperors and Elites in Rome after the Vandal Sack of 455." *Antiquité Tardive* 25: 243-62.