Both players were in the wars during the European Champions Cup win over French side La Rochelle on Sunday afternoon.
Daly is to miss up to 12 weeks with an ankle injury, while Hughes has suffered knee ligament damage which will likely take around 10 weeks to recover from.
"Elliot has injured his ankle - he has a syndesmosis injury, which is pretty common," Wasps director of rugby Dai Young said. "He is going to see the specialist to see whether any surgical intervention is required, which depending on the grade of the injury hopefully it won't be.
"Nathan has an MCL injury to his knee. Again, we're waiting on the specialist to let us know what is required, but we can safely say even the best picture, which is a grade one tear, will require a minimum of four weeks."
England take on Italy in their opening Six Nations fixture on February 4.