I prefer Greek. Sometimes translations make a big difference, not here.
Anaphoric & "Biblical" hermeneutics...
Peter's reference to "them" is called an "anaphoric" reference (as opposed to cataphoric) because it references something that happened earlier. Because he speaks of them as Gentiles and Cornelius would be the only so far in this same writing by Luke, a "Biblical Theology" (the name for this hermeneutical method) would say that surely Peter is talking about Cornelius.
When Peter says, "When we believed," he says specifically "in the Lord Jesus Christ", not, "when we believed and received the Holy Spirit." They all "believed in Jesus Christ" at different times, but they all had. The Greek does not specify any point in time for any one person's belief, nor do main English translations.
Putting it together
The "we" refers to Peter and other Jewish Christians who had believed in Jesus at their different respective points in time, those at the Day of Pentecost having received the Holy Spirit because of whenever they believed Jesus before.
The "they" most likely refers to Cornelius, who both believed in Jesus and received the Holy Spirit baptism about the same time because the Holy Spirit had been given by that point in time.
The main point via silence
That said, hermeneutics call us to let the clear things remain clear, the vague things remain vague, the anaphoric and cataphoric remain such, and the unstated remain unstated.
So, the main point includes that "they" does not specifically specify Cornelius... because, though it "Biblically" must be Cornelius, this isn't about Cornelius; the main point is what Peter says outright: To have different privileged access to Jesus and the Holy Spirit based on ethnicity would interfere with God's work.