It is a long-standing tradition that the new Prime Minister calls the leaders of the devolved nations on the first day of their premiership.
Truss won the Conservative party leadership contest on September 6.
Addressing the situation in Plenary on Wednesday (September 21), Drakeford said that he "hoped that it will be possible to conduct relationships between the UK Government and the other nations of the United Kingdom in that spirit of mutual respect", but that "there has been no opportunity as yet to test the appetite of the new Prime Minister for such an approach."
Truss did meet the First Minister briefly at Llandaff Cathedral on the day King Charles made his first visit to Wales.
Three of the previous UK Prime Ministers - Boris Johnson, Theresa May and David Cameron - all rang Wales' First Minister on the day they took on the new position.
Drakeford added: "On the day that he was appointed the then Prime Minister, Boris Johnson, telephoned me and telephoned the First Minister of Scotland, and the same thing happened on the first day that Theresa May was appointed and the day that David Cameron was appointed.
"Now, I understand that the new Prime Minister has not had the first week that she would have expected and that business as usual has not been characteristic of recent days, and now the Prime Minister has left the country.
"But, I hope that it will not long be delayed before she does find an opportunity to speak with the elected leaders of other Parliaments in the United Kingdom."
During the height of the pandemic in summer 2020, Johnson did go several months without contacting Drakeford despite repeated calls from the First Minister.
It's understood that King Charles III made contact with Drakeford within 24 hours of his mother Queen Elizabeth II's death.
The UK Government has been approached for comment.