Cold emailed small PE firm — interviewed, now unsure if I’ll hear back. Follow up?

I cold emailed a small RE PE firm in early March with my CV. The MD replied quickly and seemed eager — we set up an interview in person for April 15 (2 days ago).

Had the interview— spoke with the MD, an analyst, and two directors. Got some technicals (IRR, investment case for £100m), pitched a logistics RE thesis, and tied in my RE spring week experience + property pitch competition experience which I won (I am first year btw). Felt like I crushed the technical side and showed strong interest overall.

They reviewed my CV and seemed genuinely engaged. At the end, the directors mentioned summer is usually quieter, and said they’d look into bringing me in for 2–4 weeks, possibly longer, but they’d have to figure out the logistics. There’s no formal process in place. Can't remember what they said exactly but it was unclear when they would get back to me.

I sent a LinkedIn request couple hours after interview and the MD, analyst, and director all accepted my LinkedIn request pretty much instantly. I sent a short thank-you message to the MD on LinkedIn after the interview saying I appreciated the time and enjoyed meeting the team. No reply — which I get because it doesn't require one — but now I’m just not sure what to expect.

It’s only been a 2 days, but I’m wondering when and how to follow up, since there’s no set timeline and I don’t want to miss the window. They’ve had interns before (this spring), and they seemed serious about the idea — but nothing was confirmed.

Should I follow up via email? How soon is too soon — should I wait a few days or a week? And if I do follow up, what’s the best tone to take? Additionally, I received my exam timetable and can now start my internship earlier than I initially indicated, which could give me more flexibility. Should I mention my updated availability in the follow-up email?

On top of this, I had a call with someone in insurance back in January, who promised me a role this summer, but despite following up, I’ve been ghosted. It’s making me worry that things might fall through. Any advice on how to handle both situations would be really appreciated!