Even with a full car driving licence, you cannot ride a 125cc bike because you will need the Compulsory Basic Training (CBT) certificate to do so.
This means that you will need to retake your CBT if you wish to ride the 125cc bike and that's going to be valid for another 2 years if you pass.
You may want to consider passing the motorcycle test (both theory and practical) so that you don't have to keep on doing the CBT after it expires. This is going to save you time and effort and allow you to ride your bike whenever you like without worrying about passing tests.
On a side note, a car licence issued before the 1st of February 2001 entitles you to ride a 50cc moped (max 30mph) but if you licence was issued after that date, then you will need a CBT.
Source : http://www.direct.gov.uk/