As you now have to apply for the new visas online my guess is that it will flag up (all that info is on your passport) - years ago you had to apply to the embassy for a different visa if you had certain convictions
Yes with a VISA. The VWP is now online but that has no access to British criminal records. Criminal records are not on passports. Officially you are supposed to apply for a visa if you have even been arrested for what the septics call a crime of moral turpitude. Depends on the crime but they probably have no way of knowing at resent but if they do find out then your feet won't touch