To provide a list of airports, hospitals, and churches near Catford, Southeast London, I’ll need to make some assumptions about the location, as "The Catford Conundrum" seems to refer to a specific topic or article rather than a precise geographical location. Catford is a district in Southeast London, and I’ll base my answers on this area.

Airports near Catford, Southeast London:

1. London Heathrow Airport (LHR) – Approximately 17 miles from Catford.
2. London Gatwick Airport (LGW) – Approximately 20 miles from Catford.
3. London City Airport (LCY) – Approximately 8 miles from Catford.
4. Biggin Hill Airport (BQH) – Approximately 10 miles from Catford.

Hospitals near Catford, Southeast London:

1. King’s College Hospital – Located in Denmark Hill, approximately 4 miles from Catford.
2. University Hospital Lewisham – Located in Lewisham, approximately 2 miles from Catford.
3. Princess Royal University Hospital – Located in Farnborough, approximately 7 miles from Catford.
4. Queen Elizabeth Hospital – Located in Woolwich, approximately 6 miles from Catford.

Churches near Catford, Southeast London:

1. St. Laurence Church – Located in Catford, a historic church serving the local community.
2. Holy Trinity Church – Located in Catford, another significant religious landmark in the area.
3. St. George’s Church – Located in Catford, serving the local community with various religious and community services.
4. All Saints Church – While there are several All Saints churches in London, one is located in nearby Peckham, but for Catford specifically, there might not be an exact match. However, churches like St. Andrew’s Church in Catford could be an alternative.

Please note, the exact distances and listings might vary based on the current and most accurate geographical data available. These answers are provided based on the general knowledge up to my last update in 2023 and might not reflect any changes or updates that have occurred after that date.