The problem is that I compiled Poco for both Linux and Android. Firstly, I compiled for Linux:
./configure --config=linux --no-tests --no-samples
make -s -j4
sudo make install
I then repeated this for Android:
./configure --config=android --no-tests --no-samples --prefix=/usr/local/android
make -s -j4
sudo make install
As I hadn't done a make clean after for the Linux config, the make install for Android config found (and copied) the compiled libs and both Linux and Android.
Unfortunately Qt then linked to the wrong libs, which explains the error message I got.