HMRC announced a change on their end in the below announcement; SSLv3 will no longer be supported, TLS will be used instead.
More information on the gov.uk website, as advised in the announcement: https://www.gov.uk/government/collections/hm-revenue-and-customs-service-availability-and-issues
The below advisory Note has now been released:
Please also note that the below change has recently been announced:
Government Gateway LIVE (Production) environment at https://secure.gateway.gov.uk - SSL will be changed to TLS Wednesday 6 January 2016 between 10:00 and 11:00 GMT.
Please make sure you've carried out the transition until this date. The above Note has been updated with this information.