Bellamy, who talked extensively about his long battle with depression last year, had been part of the Anderlecht coaching staff since 2019.
The 42-year-old has been working as former Manchester City team-mate Kompany's right-hand man since earlier this year, but he will officially step aside on Monday.
A statement on Anderlecht's website on Sunday read: "Assistant coach Craig Bellamy will leave the coaching staff of RSC Anderlecht on Monday.
"Bellamy already struggled with mental problems in the past and these have resurfaced again recently.
"With his departure, the Brit took a very difficult, but absolutely necessary decision concerning his health.
"Bellamy said goodbye to the players in the dressing room on Sunday afternoon after the 7-2 victory against KV Mechelen."
Former Wales international Bellamy retired from football in 2014 following a second spell with hometown club Cardiff City.
He moved into coaching with Cardiff at academy level before departing in January 2019 and taking up a similar position with Anderlecht.
Following the news of Bellamy's departure, Kompany added: "Craig has long struggled with his mental health. Now the monster is back there.
"Health is above football. We are obligated to give him the chance to get well."
It is Bellamy's first senior managerial role, with the former Wales captain signing a four-year contract.
As a player, he made 78 appearances for the national team between 1998 and 2014, scoring 19 goals.
"It's an incredible honour for me to be given the opportunity to lead my country, and it's the proudest moment of my career," Bellamy told the FAW website. "It was always my ultimate dream to become the Cymru head coach, and I am ready for the challenge."
Bellamy had been named as Burnley's acting head coach following Vincent Kompany's move to Bayern Munich in May, but turned down the opportunity to stay as a coach following Scott Parker's appointment.
His first match in charge will be at home against Turkiye in the Nations League on September 6.
Bellamy was named the new Wales boss on Tuesday following Rob Page's sacking last month, with the country having failed to qualify for Euro 2024.
It is Bellamy's first senior managerial role and the former Wales captain has signed a four-year contract. As a player, he made 78 appearances for the national team, scoring 19 goals.
Bellamy had been named as Burnley's acting head coach following Vincent Kompany's move to Bayern Munich in May, but turned down the opportunity to stay on as a coach following Scott Parker's appointment.
He called taking the Wales position "my ultimate dream" when the deal was agreed and then faced the media for the first time on Wednesday, explaining why he had opted not to remain with Burnley.
"Timing in football, you don’t always get that luxury," Bellamy said. "The last three months it became clear to me I was happy but I needed more.
"I needed to look for a number one spot for me, as a person for my progression and obviously didn't know what that [opportunity] was going to be at that time.
"I just sort of made it clear to myself that this is what it's going to be and this is how I'm going to go about it. Then obviously the situation changed here with Wales.
"Wales has always been really important to me. I've spent a lot of my career away from here as well but I've had certain periods where I've been here and I was born here.
"The opportunity to lead your national team is rare and [it is even more rare] to get it as a first opportunity.
"It became clear to me that if there was an opportunity to do this, this is the one I wanted and I'm grateful."
The first match in charge for ex-Liverpool striker Bellamy, who says he has learned hugely from his time at Burnley, will be at home against Turkiye in the Nations League on September 6.
"Playing wise, I do like front-foot football and I like pressing," he said. "The team comes first so we are going to be difficult to break through. We build from the back. We commit.
"I want to dominate in every aspect and win football matches.
"I just had two seasons, the first winning nearly every week [in the Championship], then going to the Premier League and losing most weeks. I’ve seen the level and how much you learn. You learn more when you’re losing.
"In order to qualify for tournaments, with the detail you have to go into, the work starts now."
