The DVLA is experiencing delays with paper applications owing to a large number of enquiries and also staffing issues relating to the COVID-19 pandemic.
As a result, MPs offices have received a lot of complaints from constituents worried about their applications and whether or not they are still permitted to drive whilst they are waiting for them to be processed.
The DVLA has a useful page which shows their processing times and contains information about eligibility to drive here: https://www.gov.uk/guidance/dvla-coronavirus-covid-19-update This page is updated weekly.
MPs have access to a hotline, the details of which can be found on the Hotlines List on the intranet. Further information here: https://w4mp.org/w4mp/w4mp/hotline-numbers-for-mps-meps-msps-ams-and-their-staff/