The registration processes in various jurisdictions are not interconnected in any way. Therefore, in general, approval of your mark in one jurisdiction does not impact the timeline of registrations in other jurisdictions.
There's, however, one benefit we'd like to point out, even if it's not directly related to the ability to register a trademark. If you decide to file a trademark application identical to the one you filed in a different country less than six months prior, you can claim a so-called "priority right". This way, your second application can share the filing date with the first application, which effectively means your protection in the second country will start dating from a sooner date.
In other words, let's say you filled your EU mark on the 1st of January and an identical mark in the UK on the 13th of March. Using the priority right, your UK application will have the 1st of January as the filing date.