A Loki physically exists in both the TVA and the Sacred Timeline. Remember we see the branched timeline that the Loki variant has come from getting reset. That means that the Loki in the Sacred Timeline carries on as normal and has no memory or knowledge of the Loki from the branched timeline because from his perspective it never happens.
Miss Minutes: But sometimes, people like you veer off the path the Time-Keepers created. We call those Variants. Maybe you started an uprising, or were just late for work. Whatever it was, stepping off your path created a nexus event, which, left unchecked, could branch off into madness, leading to another multiversal war. But, don’t worry, to make sure that doesn’t happen, the Time-Keepers created the TVA and all its incredible workers. The TVA has stepped in to fix your mistake and set time back on its predetermined path. Now that your actions have left you without a place on the timeline, you must stand trial for your offenses. So sit tight, and we’ll get you in front of a judge in no time.
Loki, Season 1 Episode 1, "Glorious Purpose"
This also means the consciousnesses are unlikely to be linked. We also see another Loki variant, Sylvie, and the Loki variant that we follow in the show has no link to or knowledge of her. If there was a link or consciousness shared you'd think they would share it too.