Can you be a commercial pilot if you have a criminal record?
British Airways has this to say:


  If you have been cautioned, warned or convicted of a criminal offence (other than parking offences) we suggest you check whether you will be able to obtain an airside pass by visiting www.dft.gov.uk.
  
  In order to be employed as a pilot with British Airways you will be required to undergo a Criminal Record Check for every country in which you have been a resident, or worked in, for 6 months or more in the past 3 years.


Admittedly, the DFT's website is a maze of information, and it seems very difficult to find any information; I'd suggest giving them a call, to find out for your particular convictions.

Airside Manager has some more detailed information on airside passes, in general.