I'm in the process of writing an Email to Mark Serwotka head of the PCS Union whose members at the HMPO are taking this industrial action. I'm going along the lines of HRA 1998 Freedom of Movement and Art 8 Rights to private family life being affected by this strike.
You cannot travel internationally whilst your passport is in the process of renewal.
You would have received confirmation of this in writing from VFS Global or the agent you used.
If you have to leave Thailand you will be able to apply for an Emergency Travel Document (Passport) from the embassy that will allow you to travel back to the UK only.
If you used an agent maybe ask them to chase up the HMPO.
Speak with the company HR department. Contact the Immigration office in BKK to verify what documentation you require.
Gather all the documentation you have to support your application (translated if required) or arrange for it to be sent / completed (i.e Police checks from country of your citizenship if applicable) and together with the company HR representative attend Immigration in-person as you are in country.