Have we register with US Embassy?

I am sorry for this question (I guess, it was discussed before but I could not find it): when US Citizens are coming for permanent living in UK, have they register with US Embassy as expats? In my case, there is no a sponsor who could register me.

I'm not quite sure I understand what you're asking.

If you're asking do you have to register, then the answer is no.

If you would like to register, you can do so here: https://travelregistration.state.gov/ibrs/

So, when time to fill in Tax return comes, can I use the IRS service (like an H&R Block) via Embassy not being registered?

I think this is in the wrong forum, as the OP is talking about moving from the US to the UK.

So, when time to fill in Tax return comes, can I use the IRS service (like an H&R Block) via Embassy not being registered?

Yes.  You don't have to register with the US embassy when you arrive in the UK.  The only thing I did was register to vote.  I paid my US taxes using my US accountant. 

Thank you, Kristi. I am fine now.

For others who might be interested in the same topic, I have found related Web site: https://travelregistration.state.gov/ibrs/