Avoiding the bug
About the Forsworn Conspiracy quest:
Followers may get permanently stuck in the Shrine of Talos,
regardless of if the quest is completed and you escaped from Cidhna
Mine. If this happens, the follower will not leave even after you tell
them to go home, and if they are moved elsewhere via console, they
will walk back to the Shrine of Talos, no matter where in Skyrim they
are.
From: UESP Wiki, "The Forsworn Conspiracy - Bugs"
I'd recommend not having a follower while doing this quest to avoid them getting stuck in the Shrine of Talos. Or you could ask them to wait outside the Shrine. Multiple backup saves are also recommended.
Fixing the bug
If you've already encountered the bug and can't load a previous save, try these*:
Try the enable and disable console commands.
Make a save game first, and then in the game, open up the console by pressing ~, then click on your follower (in OP's case, Aela). Aela's RefID (0001A697) should show up in the screen. If the RefID is correct (meaning you have selected Aela correctly, and not another object), type disable and then enable in the console. This should 'reset' her.
If Aela is not near your location, you can still select her by entering the console command prid 1A697, after that type disable and then enable. Then try the moveto player command (which moves that prid-targetted entity near the vicinity of the player) and see if she gets 'reset' and decides to stop her pilgrimage to the Shrine of Talos.
If it still doesn't work, try finishing the Forsworn Conspiracy quest and the next quest that it will trigger, No One Escapes Cidhna Mine. See if your follower will now follow you.
(I know OP has already posted a solution that worked for them. Just posting alternative solutions in case someone else experiences a similar issue.)
It is possible to remarry if you use the console as it is explained here: http://www.uesp.net/wiki/Skyrim:Marriage
Select spouse.
Enter the following console commands:
removefac 51596
player.removefac C6472
resetquest 74793
resetquest 21382
setstage 74793 10
This should make you able to marry again after 24h (whether you've installed Dawnguard or not)
Best Answer
Billy Mailman's answer is rather complete, but just simplifying things, the following worked well for me:
If you are playing on a computer:
prid 02002b74and hit enter to select SeranaSetpqv DLC1NPCMentalModel Lockedin_var FalseThis will cause the "Follow me" dialogue option to reappear.
This solution was found on this link.