WordPress – NAT gateway set up doesn’t work

I have implemented an environment for a WordPress website. I have a loadbalncer (ALB) that is connected to an EC2 server plus an AutoScaling group(for failover).

As we need to update WordPress and install some plugins on EC2 server, I need to set up NAT gateway to allow internet connection from wordpress server(ec2) to internet.

I have set up the NAT gateway like the following link:
https://aws.amazon.com/premiumsupport/knowledge-center/nat-gateway-vpc-private-subnet/

However I could not be able to reach the Website (DNS of ALB) afterward.

My Routetable set up looks like below:

                      Route                Subnets
Main Route Table        10.0.0.0/16 local       -
                     0.0.0.0/0   natgw

Public Route Table        10.0.0.0/16 local       10.0.10.0/24, 10.0.11.0/24  (ELB subnets)
                     0.0.0.0/0   igw

Private route B      10.0.0.0/16 local       10.0.20.0/24,10.0.30.0/24 (EC2 subnets)
                     0.0.0.0/0   natgw  

Private route C      10.0.0.0/16 local       10.0.21.0/24,10.0.31.0/24 (RDS subnets)
                     0.0.0.0/0   natgw

The Public route is assigned to two public subnets which are connected to the LoadBalancer.

The Private route B,C are assigned to private subnets: Web server()EC2 plus database-RDS.

Any help would be appreciated.