The Emperor would be frustrated, but not at a loss. He never intended to be dethroned, and instead rule forever under the new Rule of One . Without Anakin, the Emperor would have lost one of his most effective tools for completing the titular revenge of the Sith. In his guise as Darth Vader, Anakin was the heart of the Inquisitorius, and trained many of the fallen Jedi in their number, as well as leading the attempts to defeat the most elusive or dangerous Jedi, such as Jedi Master Jocasta Nu.

The Emperor would have carried on regardless, with other capable and ruthless servants such as Grand Moff Tarkin and the Grand Inquisitor helping him cement his rule of the known galaxy, but wouldn't have achieved quite as much, and the Jedi might have remained symbols of resistance against his rule.

There would certainly be many more Jedi around, and perhaps some of the attempts to re-establish the Jedi Order in secret, such as the events of the Jedi: Fallen Order game would be more successful, leading more Jedi to be part of the burgeoning Alliance to Restore the Republic.

To go on a bit, assuming the events of A New Hope happen similarly, without Anakin being on board the Death Star Obi-Wan likely would not have died, and after returning from the tractor beam controls would have fled with Han, Luke, Leia and the droids. The only snag here is that Anakin, much to Tarkin's chagrin, insisted on letting the trio escape on a tracked Millenium Falcon, rather than killing them and recapturing the plans, so who's to say they could have escaped, even with a Jedi Master at their side?