If you change your DVD player to be region free and you get a DVD that is region 2 will it play or will the DVD have to also be region free?

Europe is region 2 for DVD and Bluray disc encoding. Most region 2 discs will play without a problem in a UK player.

Note that there is a facility to add a sub region to the code, so dividing Europe into smaller regions. It is rarely, if ever used, but technically, it is possible to further restrict the use of discs from the remainder of Europe in the UK.