On the final days of the World Wars, I imagine every man and woman who were enlisted in the British Army were sent home to their families. Is there any record of the amount of soldiers who remained enlisted after the fact, or what the process was for declaring their exit from the army?
I can't answer for WWII at all but for WWI there is a good write-up and links to the service records databases located here.
The short version though is that, due to the bombing of London during WWII, many of the service records were lost but some were recovered from the destruction - such records being labelled the "Burned Records". As a result, something like 60% of all service records were lost which makes answering your question quite difficult.
However, as per the link above, you can search the database via a subscription fee if you have specific questions.
Now, British soldiers enlisted "for the duration" of the war and, due to the pressure both to keep to that promise and also to release the soldiers back into the workforce, most of the soldiers were automatically discharged back into civilian life.
However, some offers were made to those who met certain conditions. Here is a brief summary. It should be noted that there was significant financial incentive to re-enlist and that many did take up the opportunity.
Unfortunately, I am not aware of any comprehensive statistics for the rate of re-enlistment.