CE680 Captain reported receiving a TAWS warning while on a circling approach resulted in a CFTT event. Flight crew corrected and continued approach.

Narrative

ZZZ was running the ILS XX Circle to Runway XY approach. Winds were favoring Runway XY; and while not particularly strong (I don't remember exact METAR but something around 10kts out of the NNE with maybe small gusts) wind at 1500 ft was about a 20 kt push away from the airport and a smaller push towards the Runway XY final approach course). The Second in Command (SIC) and I were on our second flight together; and first flight with him flying. I have done the circle to XY quite a few times in my career; but it had been a long time with since I've done it with this company. ZZZ doesn't seem to want to use that approach nearly as much since an accident; seeming to prefer straight-in to Runway XX; so I felt somewhat rusty doing it. We did a thorough brief; including discussing the threats of likely being asked to keep speed up until ZZZZZ and the idea of turning just inside the building as well as the taller towers to the south of the building. During the approach the SIC did a great job configuring and slowing once we reached ZZZZZ (we were assigned 170kts until ZZZZZ). He correctly started the circle at ZZZZZ1 as briefed. However; when he turned he turned somewhat too far to the right; in my opinion. The 20 kt wind began to push us further from the airport and it became obvious we were drifting towards the far side of the building. I think I let this occur too long without issuing a correction; and about the time I was speaking up to say 'lets turn another 20 degrees left' the TAWS system gave us a 'caution; obstacle.' At that time I reached up and dialed in an immediate left turn. Part of the reason I delayed may have been that I had the towers in sight the entire time; and I knew we were safe although likely outside the circling area. I shouldn't have been okay with that developing that long. We didn't receive a TAWS warning; and I think the situation was corrected quickly; but I shouldn't have allowed it to develop that far. Suggestion: As above; I think the threats and plan were well-briefed. The SIC at one point hinted that he wasn't sure if he was heading the right direction by saying 'are you okay with this?' or something to that effect. I should have more directly answered him and said 'you should turn about another 20 degrees left and aim for the inside of the building.' Also I wasn't as confident in the sight picture myself because it had been quite some time since executing this approach. The accident is always back of mind when doing this one; so the overall focus may have been too much on not being blown through the final and on giving us extra room for the final while not enough focus was on hitting a specific point on the final relative to the airport. It was a good learning experience for both of us. I need to be more assertive at times as Captain; it's not always sufficient to do a good briefing sometimes corrections need to be issued further in advance.