Both the Irish and British governments have published information on what the implications may be to pet movement from/to the UK post Brexit.

Because there is so much uncertainty surrounding Brexit still (late 2019), I would guess that we just have to wait and see how things pan out over the coming months and years, and pay close attention to the rules for this topic as they are adjusted. There are bound to be changes. For now, it appears as though little will change. The material from the Dept of Agriculture indicates that "pet travel rules for entry into the UK may not change for some time after Brexit". Who knows what those change may be, or even what "some time" means. We will just have to update each other as we learn more.

In the meantime I wanted to share with you some links to official information that you can peruse:
The Irish Deparment of Agriculture informaiton have created a resource on pet travel post Brexit which you can find here

Likewise, the UK have published information on the topic here

I plan to update this post as more info becomes available.

brexit dog credit: segrott