The driver of the Edward Coaches bus and a backup driver were among those in hospital following the accident on Friday that occurred near Sursee, the ATS news agency reported.
The driver of a Swiss tow truck was also injured, Lucerne cantonal police said.
Jason Edwards, a member of the board of Edward Coaches, a family-owned business based in Wales, said the seriously injured victims will have to remain in hospital for several more days, ATS reported.
Their lives are not in danger but they suffered fractured back bones, the news agency said.
Thirty-four passengers on the bus, who suffered minor injuries, were able to return to Wales on Sunday.
The bus had been travelling from Italy when the crash occurred shortly after 12.15pm after the coach passed through a tunnel on the motorway.
A truck from Poland and another from Slovakia were involved in the accident, which remains under investigation by Lucerne prosecutors.
Twenty ambulances and three helicopters responded to the scene along with firefighters, police said in a statement.
Estimated damage totalled several hundred thousand francs, the statement said.
The nationality of the victims was not immediately known, police said.