Assuming you replace the card before the end of 2020, then you will be treated in the same way as an EU citizen in Germany, as the Brexit transition phase is still in place. So the process is fairly simple.
When you apply to change your licence then the German authorities will contact the DVLA in the UK to validate various details about the card and yourself and then, if everything is in order, issue you with a new German driving licence in place of the UK licence.
The German authorities will not care if the address stated on the card was correct at the time when you left the UK or not. This is a matter for the UK authorities.
You are in fact not required to replace your UK licence for a German one until the UK one expires (which is normally 10 years after issue).
So this means that any address details printed on the card could anyway be up to 10 years out of date.