Rhys19 SQLSTATE[HY000] [2002] Connection timed out
There can be two causes for this: a) no connection could be set up at all or b) the connection dropped after a while.
Testing that you are able to connect from your remote machine manually would help:
mysql -h<hostname or ip> -p3306
It is very likely that your home connection is too slow for it to respond in time. I also recommend against hosting your database like this, performance will be drastically impacted.