The already-limited early services described above get thinner still at weekends and on public holidays — the Piccadilly line's later Sunday start is one example, but reduced frequency and engineering works are common across the network on Saturday nights and Sunday mornings specifically, which is exactly when Night Tube and weekend engineering closures overlap. A journey that is merely tight on a Tuesday morning can become genuinely unworkable on a Sunday at the same hour, so it is worth checking TfL's own service status specifically for your travel day rather than assuming a weekday timetable applies.
This is one of the clearest cases where booking a transfer the night before, rather than trying to plan a public transport route on the morning itself, actually saves time as well as stress: TfL's live service updates are not published far enough in advance to plan around reliably, whereas a pre-booked pickup time is fixed regardless of what the transport network is doing that particular morning. For a flight that absolutely cannot be missed, that certainty is the whole point.