Can a male English bulldog and a female Basset Hound have puppies on their own?

Yes, this is possible, but the Bulldog would likely need human assistance as they have lost most of their capability to mate independently. Matings between two Bulldogs need humans on hand because the Bulldog breed can no longer mate without assistance, nor give birth without a caesarian. These problems may be present in a mating with a Basset Hound.