Bhagiratha is the biological son of Dilipa. In this chapter of the Bala Kanda of the Valmiki Ramayana, during Rama and Sita's wedding the sage Vasishtha describes the ancestry of Rama. In particular, he says this:
From Sagara it is Asamanja and from Asamanja it is Amshuman, and from Amshuman it is Diliipa, and the son of Diliipa is Bhageeratha.
This chapter of the Vishnu Purana says the same thing:
The son of Anśumat was Dilípa 5; his son was Bhagíratha, who brought Gangá down to earth, whence she is called Bhágirathí.
But for definitive evidence that Bhagiratha was Dilipa's biological son and not someone born from Dilipa's widows after Dilipa's death, this chapter of the Bala Kanda of the Ramayana describes Dilipa actually handing over the throne to Bhagiratha while he's still alive:
That best one among men, namely king Dileepa, on anointing his son Bhageeratha in the kingdom went to the abode of Indra, namely the heaven, only by his self-acquired merits of deeds.
This is confirmed in this chapter of the Vana Parva of the Mahabharata:
And a son was born to him, known by the name of Bhagiratha beauteous, and devoted to a virtuous life, and truthful, and free from feelings of malice. And Dilipa appointed him as king, and betook himself to the forest life.
According to this journal paper, the earliest sources that mention Bhagiratha being born from two widdows are the Krittivasi Ramayana and a Bengali manuscript of the Padma Purana, both dating to the 14th century. So I think it's pretty clear that this is not an authentic story.
One note of caution: there is a famous story, told in this excerpt from the Padma Purana and popularized by Kalidasa's poem Raghuvamsa, of a king named Dilipa who propitiated the cow Nandini and thereby obtained a son, Raghu. But this chapter of the Vishnu Purana makes clear that this is not the famous Dilipa who lived a few generations after Sagara, but rather another Dilipa who lived a few generations before Rama. So don't make the mistake of thinking that Bhagiratha and Raghu were brothers; Bhagiratha was an ancient ancestor of Raghu.