We just did the same thing, somehow it all worked out.

Here's what we did

We negotiated with the buyers of our house to have access until 4pm
Settlement on our house was at 11.30
Settlement at the new house was 12.00
The removalists came at 9.30 and started packing
We had a few hiccups with the removalists, the truck was too small and broke down but we got there.