Since you are not a Templar what could they do if you told Elea?
Sweet F...King nothing
Kill her themselves?
Stand by while someone else kills her, not giving a shit?
They've already stated people that find out about Neverwhere are often killed by other agencies, and Templars don't care one bit about that, even if it's innocents.
That in mind, if Ceolag gives an option, I'm telling Elea everything, every last thing (well, almost everything. She doesn't need to know details of what happened by the lake with Finnabair).
She's already investigating stuff that will get here there eventually, might as well give her all the info that might potentially save her life in the future, instead of slowly stumbling in the dark.
If that creates even larger rift with family and templars (Kaija included), all the sweeter.