www programming

you have not read instructions where it is specified that you have to setup the path of the phpBB cookie, to be available overall the domain, so you need to setup it as
/
a single slash
AND NOT
/forum/
like it is now

Done this, may reset all session to all users into acp, so they'll be forced to relogin

NOTE: if cookie setting changed to be available over all domain because it was not, then all phpBB users may need to re-login using new board’s cookie settings: in this case, you may would like to force all phpBB users to re-login in case they saved their session cookie with a ‘remember me login’. This can be done by accessing phpBB ACP main page: check for option
Purge all sessions
Purge all sessions. This will log out all users by truncating the session table.
Click the Run now button.
Be sure that the cookie setting is correct before to execute these steps, or you’ll be logged out by your board until you’ll not reset to the correct cookie domain!

as explained and linked into install procedure:
https://www.axew3.com/w3/2016/02/config ... ll-domain/

In fact, I missed it even though I read about it, the current connection looks ok, but there are still a few problems
I attach a video of how it looks http://flo.micromedia.pl/bridge.mp4

I have a problem with iframe height - it's fixed height Redirection from the forum admin panel does not work, links from the last topics widget also do not work and opening the link in a new window does not work, message: Something goes wrong with your URL request, please leave this page.

After enabling the w3all plugin, I stop seeing the list of users http://flo.micromedia.pl/florysci/, in the original it looks like this https://ibb.co/LdBNB3Q (was the transfer of users from phpbb to wp)

have you encountered such cases?

I have a problem with iframe height - it's fixed height

why never you modified adding fixed height to iframe element?
You should in case change the heightcalculation method.

Redirection from the forum admin panel does not work, links from the last topics widget also do not work and opening the link in a new window does not work, message: Something goes wrong with your URL request, please leave this page.

of course it do not work, you choose as fancy url a single letter i
which is reserved wordpress variable, and used into page forum to detect a var i.
You can't use i, nor any of other reserved vars names.

After enabling the w3all plugin, I stop seeing the list of users http://flo.micromedia.pl/florysci/, in the original it looks like this https://ibb.co/LdBNB3Q (was the transfer of users from phpbb to wp)

into original, if wordpress users where ok about this list, there is no reason why after, if you transfer users from phpBB to wp or viceversa, it should not work. In fact an existent wp user, isn't transferred into wordpress, if exists, and nothing changed/updated, except password in case.
Instead, may the question is another: into a wordpress, i transferred phpBB users, and i expected to see them into this list, which is generated by what? An external plugin? May an external plugin that to work, add data in some way about users, but not when a wp_insert_user action executed?

p.s about heightcaluculation method:
reset the line into page-forum you wrongly edited adding a fixed height as value for the iframe (beside the fact that it will never work because overwritten by js resizer library) into what was, or rebuild page-forum,
then follow this instead:
viewtopic.php?f=2&t=1511

thanks your tips are very useful, i'm one step ahead, i am currently having another problem

I log in via phpbb - I'm logged in to phpbb and wp - it's OK

I log in via phpbb and log out via wp - it's OK

I log in via wp and there is a problem ("Incorrect Username / Password.")

I log out via phpbb (effectively) and wordpress does not log me out and the forum page is looped

i've not understand how register, setup me an account to test please

p.s
about password:
have this user try to reset password once, into phpBB or wp?
The loop problem into iframe instead seem to be due to some wrong setting that should lead to not correctly read or detect that the user isn't logged or is logged, because i think maybe, js code fail due to some reason, for this i asked you an account to test a logged in user