There are many coaches in the world who can work in England and they are not working in any team. Maye Klopp could start working there but he said he is tired and he need some rest which means he is out. Tuchel has not had a good performance recently and some other coaches like Pochettino and Potter can have more chances to start working in England as the next coach.
I think Pochettino can be the best option they can have because he had a good performance at Chelsea.
Southgate failed to bring England to be the champion for the second time and it was very painful because they reached the final with a lot of struggle. From what I saw England did not have a dangerous striker and Harry Kane did not show the class as one of the deadly strikers. After the competition ended, Southgate resigned as coach and I don't know the exact reason but what I saw was because he always failed to give his best for England.
Of the three names you mentioned, I am more interested in Jurgen Klopp than Tuchel or Pochettino. But for Jurgen Klopp, it might take time to convince himself after leaving Liverpool. Preparation after preparation must be done so that England can prepare some new strengths for the future.
Southgate should have done that because he has failed and in my opinion there is nothing missing from the England squad, they have everything but Southgate has not succeeded in making this team even better.
And overall, I think England's failure to win the Euro title is appropriate, their performance has been very bad since the start of the Euro tournament, they were just a little lucky to reach the final.
Now it's not just these 3 names but there are also several other names that are being rumored, but in my opinion, finding a coach who excels when coaching a team and has never coached a national team will be quite difficult to recruit.