Cockerill, 52, will join French Top 14 club Montpellier as forwards coach.
Wigglesworth, the interim head coach at Leicester, will be assistant to England boss Steve Borthwick after working under him with the Tigers.
Walters will be in charge of strength and conditioning heading into the World Cup in France, which starts in September.
Former Leicester and Edinburgh boss Cockerill was the only member of Eddie Jones’ backroom staff still involved in the England set-up after Borthwick replaced the Australian in December.
But his role was reduced from being forwards coach to overseeing the scrum.
Wigglesworth and Walters are the latest members of Borthwick’s coaching team at Leicester to join his England staff, with the former Red Rose captain taking defence chief Kevin Sinfield with him when he accepted the England job.
Cockerill was previously head coach of Toulon in France.
“Having coached and played in France previously, my family and I always had aspirations to return. This opportunity presented itself some time ago and it was too hard to turn down,” Cockerill said.
Cockerill will replace Olivier Azam at Montpellier, under head coach Philippe Saint-Andre who welcomed the recruitment of a “world class coach”.
Borthwick’s reign as England boss started with a third successive defeat by Scotland in their Six Nations opener on Saturday.
England continue their campaign at home to Italy on Sunday.
Jones cleared to play for Wales against Scotland
Wales second row Alun Wyn Jones will be available to face Scotland in the Six Nations tie at Murrayfield on Saturday, despite coach Warren Gatland having ruled him out after the 34-10 loss to Ireland in Cardiff on Saturday.Gatland told reporters Jones had failed a Head Injury Assessment, which would see him miss the trip to Edinburgh.
But subsequent tests which are more detailed have cleared the experienced lock and at this stage he has no signs of concussion, team officials confirmed yesterday.
Team doctors have been in contact with World Rugby to keep them abreast of the situation.
There may be more good news for Wales as prop Tomas Francis has responded well to treatment for a calf problem and looks set to be available.
Gatland insisted there were positives to take from the heavy loss to the Irish, in which Wales conceded nine penalties and three tries in the opening 20 minutes, a disastrous start from which they never recovered.
In particular he praised young centre Joe Hawkins, 20, who earned his second cap and was one of the most impressive performers from either side.
“That’s what you want, a bit of backing from the head coach,” Hawkins told reporters yesterday. “It’s about not trying to be a different player, just go out there, and be confident in your own thing.
“It’s surreal being out on the training field with likes of AWJ (Jones) and Tips (Justin Tipuric). I’m trying to take in all their experience and then it’s about putting it into your own game.”
