United will receive £100,000 in compensation for Laws, as well as an undisclosed fee for Wilcox.
Adkins will take over first-team duties with immediate effect, with his first test being Friday's FA Cup first round trip to Cheltenham Town.
He will be assisted by veteran players Andy Crosby and Ian Baraclough, with youth team coach Tony Daws taking charge of the reserve team.
"Adkins has nine years of experience of the club and is well respected by all the players," said Wharton, who revealed he offered Laws a one-off financial incentive to stay.
"I have the utmost respect for him, he is totally focused on the well being of Scunthorpe United and I trust him implicity.
"I spoke to captain Crosby and Baraclough, who are both very experienced and influential players. They fully supported my decision and agreed to work to ensure as little disruption as possible."
Wharton must now begin a search for Laws' full-time replacement, but insists there is no rush.
"A number of possibilities have gone through my mind, but I am solely concerned with maintaining our current form," he added.
"I am confident that the temporary arrangements will give us the best chance of achieving just that.
"In the next few weeks we will monitor our progress and consider all the options available to us."