The last District line train from Earls Court to Southfields on Mon/Sat leaves Earls Court at 0053. So we need to find out what you can do to connect with that.
The last Piccadilly line train from Heathrow T123 that goes through to Earls Court is at 2345 (T4 2335. T5 2342). There is another train from T123 at 0003 (T5 0000, no T4 option) but it terminates at Hammersmith. In principle there's a District Line train from there to Earls Court, but it arrives at ECt at 0053 so you may not succeed in making the connection. The very last train departs Heathrow T123 at 0021 (T5 0018 T4 0009) terminating at Hammersmith. At least these would get you a rather closer to Southfields for a taxi.
Another possibility is to go via Paddington on Heathrow Express and use the direct Paddington-Southfields part of the District Line (Edgware Road- Wimbledon line). The last train from Edgware Road towards Wimbledon departs at 0020, so I imagine a call about 0021 at Paddington. I'll leave you to check Heathrow Express timings to see if that gives you any additional leverage.
In sum, you can't guarantee, at all, succeeding in making these journeys and should have a few options up your sleeve depending upon how quickly you make it through formalities.