Differrence in edit mode and on Openstreetmap

Can anyone tell me when, or how, change of street name will be visible on OSM?
New name is already available in search and is seen on map when you switch on edit mode, but on OSM there is still old, wrong name.

The cache of your browser is the problem.
Zooming in/out a few times is most of the time enough to show the new situation.
If you have chrome as your browser, then start a new incognito window to bypass the cache for the current session.
If you see the new data in your editor, you can be sure it IS there and only takes some time to show up.
For every zoom level the (visible) map has to be rendered again after you did some edits.

What the editor shows you is the real Openstreetmap. What you call Openstreetmap is just a selective approximation provided to help people edit the real map.

As well as browser caches, the rendered images can get stuck in ISP caches. Also, all but the highest zoom level are rendered in the background. For the “standard layer” they are normally only a few minutes behind but can get up to hours behind. Some of the other “layers” can be over a week behind the actual map.