The villa of Livia was located below a smallish mound between the Tiberine and Cassian roads, near the locality known as Prima Porta (then in the Roman Campagna, now one of the city's suburbs). ^^1 Antonio Nibby, one of the foremost archaeologists and antiquarians in the Papal States, had already identified some of the few structures ^^2 that could be seen emerging from the earth as belonging to a building of the Augustan era - possibly the very villa of Livia - in 1849, but he merely limited himself in reporting his findings within his Analisi storico-topografico-antiquaria della carta de' dintorni di Roma (Tipografia delle Belle Arti, Roma 1849). Perhaps that year's revolutionary uprisings may have had some kind of role in delaying further inspections, but we don't know with certainty... well, at the very least I don't!
As to why the place wasn't excavated earlier, keep in mind that the Campagna was a quite desolate and sparsely inhabitated - not to mention dangerous - place at the time; brigands and malaria weren't uncommon occurrences.
The events of 1860 may have discouraged further research/excavations as well.
So the mound laid untouched for another 14 years until a three-man party, led by a certain Giuseppe Gagliardi with the assistance of a Mr. Senni and a Mr. Ambrogi ^^3 (they'd realised the site's significance, having probably learnt about Nibby's studies) began excavating the hill in the hope of finding something. And something they did find!
G. Henzen described the findings as such:
[...] They produced, apart from many decorated floors they'd discovered almost at ground level, three very beautiful busts portraying Roman characters [...] Numerous pipes made of lead bearing the name of Titus Caesar were also discovered, thus confirming the attribution of the afore-mentioned villa [...] Various chambers, which have already been opened, showed walls once covered in lively colours [...]
Then...
[...] On the 20th [of April] one of the most interesting and beautiful [antiquities] of this kind to have ever been excavated was brought to light. This one had been found in front of the villa, on a surface facing the Tiber which might have been once a porch or a terrace located right in front of the building's façade. It is a life-sized portrayal of Augustus and is wonderfully complete in all its parts, in such a way that not even the tip of its nose is missing. The [statue's] right arm appears to be broken, and so is the lower section of its legs; but all the parts have survived, and they only need be re-attached...
Seems like they had just found your guy, OP!
[...] A few surviving bars of iron, [protruding] from the afore-mentioned arm and left leg, show that the statue had already been broken in antiquity. Its head was grafted on the torso. The emperor, portrayed when he was approximately 40 years of age, has his head slightly tilted on the right. He is nude, and so are his feet - something which, considering the armour which covers the rest of his body, lead us to believe he'd been portrayed when he had already been deified [...] or as a god, or a hero.
Sources
^^1 AA. VV. - Bullettino dell'Instituto di Corrispondenza Archeologica per l'Anno 1864 (Tipografia Tiberina, Roma 1864).
^^2 ibid.
^^3 AA. VV. - Bullettino dell'Instituto di Corrispondenza Archeologica per l'Anno 1863 (Tipografia Tiberina, Roma 1863).