unable to mount recovery points on windows server 2012

Have you tried a different browser? Or restarting the core service? Or rebooting the entire core server? It sounds like something may be hung up and a quick restart would probably resolve it.

Restarted the core earlier this morning, exhibiting same issue.

Have you tried using the local mount utility to see if it's just the core GUI or not?

There may be a background job (i.e. deferred deletes) blocking it. (Hit the little gear icon on the upper right side of the tasks page to display these jobs). Additionally, taking Tim's suggestion one step further you may want to try mounting a recovery point using powershell.
To do it, open an elevated powershell console on the core and type:
PS c:\> new-mount -protectedserver 'MyMachineName' -mounttype write -path 'c:\Mounts'
This will mount the newest recovery point. Type:
PS c:\>remove-mounts -all
to dismount all mounted recovery points.
The full info re the new-mount is here:

New-Mount -core [host name] -user [login] -password [password] -protectedserver [name | IP address] -mounttype
[read | write | readOnlyWithPreviousWrites] -drives [drive names] -volumes [volume names] -path [location] -rpn
[number | numbers] | -time [time string] -showprogress

Options:
-core Optional. Remote core host machine IP address (with an optional port number). By default,
the connection is made to the core installed on the local machine.
-user Optional. Login to the remote core host machine. If you specify a login you also have to
provide a password. If none are provided then the logged in user's credentials will be used.
-password Optional. Password to the remote core host machine. If you specify just a login then
you're prompted to enter a password. When none are provided then the logged in user's credentials will be used.
-protectedserver Protected machine with a recovery point(s) to be mounted.
-mounttype Optional. Specifies a mount mode. Available values: 'read' (read-only),
'readOnlyWithPreviousWrites' (read-only with previous writes), 'write' (writable). By default, read-only mode is
enabled.
-drives Optional. Comma-separated list of volume names to mount. This option is obsolete, use
'-volumes' instead.
-volumes Optional. List of volume names to mount. If not specified, all volumes will be mounted.
Values must be enclosed in double quotes and separated by coma. Don't use trailing slashes in volume names.
Example: "c:","d:".
-path Path to a folder on the core server where recovery point will be mounted to. Folder is
created automatically if not exists.
-rpn Optional. The sequential number of a recovery point to mount (use Get-RecoveryPoints
command to get the numbers). Specify several space-separated numbers to mount multiple recovery points with a
single command. In this case data from each recovery point will be stored in a separate child folder. Note: if
neither option -time or -rpn is specified then the most recent recovery point that successfully passed integrity
check will be mounted.
-time Optional. Determines recovery point(s) to be selected for mount. Available values:
'latest', 'passed', exact time in the format "mm/dd/yyyy hh:mm tt" (for instance, "2/24/2012 09:00 AM"). Keep in
mind to specify date time values of the time zone set on your PC. Note: if neither option -time or -rpn is
specified then the most recent recovery point that successfully passed integrity check will be mounted.
-localdrive Optional. Perform mount to user disk on local PC.
-showprogress Optional. Specify to show the mount progress to a folder on the core server.