It's 1 am here, so I shall be brief.
As with you, we have done both versions of your schedule over the years
but, on our frequent trips to England, we usually do the arrive/pick up
car/drive away from city version, and that works well for us.
Flights from Australia usually arrive in the early morning, as do yours
- around 5am or 6am for us - so the experience would be similar.
Having the early time in the UK (but out of town) often leads to us finding
out from friends, newspapers etc about things in London (exhibitions,
shows, new restaurants etc) that we may not have tripped over going to the city
directly, and so gives us the opportunity of organising them for the end
of the trip.
Shopping can be done at the last minute too, reducing the tedium of carrying
extra goods etc.
If you are in the city for the last part of your stay, you can hand back
your car, then take the express train service to Gatwick (if you do decide
to use that airport), checking in your luggage at the station which is
also a boon. Do check in 2 hours before departure though if you want to
do the luggage bit from the station.
Whatever you do will work well, I'm sure! Just a question of what works
best for you.
Happy planning - and good night!