Blue Badge apIication

All you can do is contact the Blue Badge team and ask them to update you on the progress of your application.

Their webpage https://liverpool.gov.uk/parking-roads-and-travel/parking-permits/blue-badges/apply-for-or-renew-a-blue-badge/ does say
It can take up to 12 weeks from receipt of payment for your badge to be issued. At times, delays may be even longer.