".....don't want to stick a finger up and say 'Here I am, come and sink me!'
That was Dönitz's big mistake in the Battle of the Atlantic! "
It is hard to make general statements about the technology in the Battle of the Atlantic, since to a large part it was a technological battle, and the balance between the relative capabilities of different technologies altered - sometimes on a monthly basis or less!
RDF was well understood by the Axis powers, and they were required to keep their messages short to avoid a fix. For a mechanically rotated aerial, which the Germans used, getting a fix required a fair bit of time, and getting an accurate fix could take a minute or more. Shore stations then had to transmit the information to interested parties, and so a U-Boat was fairly safe if it sent occasional short messages and kept moving.
I believe that the Royal Navy responded with the Huff-Duff dual loop antenna which you will find on
most RN ships of the period, feeding into a CRT PPI
indicator. This could give immediate accurate directions, sufficient for an attack to be mounted directly down the bearing. Though, of course, this could also be used to tempt a corvette off station to enable another boat to attack. It had originally been developed to track lightning strikes...
Lütjens, of course, sent a famous long message enabling precise identification of the Bismark, but the British then made a plotting error and sent their Task Force in the wrong direction for a while....