Is the Land Registry functioning?

Yes. The Land Registry published a new service update on 14 May, here:

https://www.gov.uk/guidance/coronavirus-covid-19-impact-on-hm-land-registrys-service

Importantly, the Land Registry will process registrations where documents have been executed using the Mercury signing approach:

For land registration purposes, a signature page will need to be signed in pen and witnessed in person (not by a video call). The signature will then need to be captured, with a scanner or a camera, to produce a PDF, JPEG or other suitable copy of the signed signature page. Each party sends a single email to their conveyancer to which is attached the final agreed copy of the document and the copy of the signed signature page.

To summarise some further points:

• Most information enquiries are experiencing minimal delays
• Registrations of new titles, such as on sales of part or new leases, and applications to update existing titles, are experiencing more significant delays but can be expedited via the expedite service
• Cancellation dates for replying to requisitions are extended until further notice
• Access to free documents on the land registry portal has been extended to 90 days from completion of the transaction
• Identity requirements have been relaxed. The Land Registry will now raise a requisition for identity documents, and not cancel applications
• Requests for extensions to a notice or objection period will be granted if lawfully possible
• Land charges searches can be submitted electronically with PDF documents