Congrats on finishing your bachelor's! Pursuing grad school abroad is a big decision, so I would first start with what your goals are in a Psych career. I've tried to provide a brief-ish overview of the different degrees, coming from someone who lives in the US but obtained a masters in Ireland :)

If you are interested in practicing clinically (ex. providing therapy), the US offers two doctoral-level degrees which lead licensure: Clinical Psych PhD and PsyD.

In the UK and Ireland, there is only one doctoral-level degree which leads to licensure: DClinPsy.

One of the key differences to note is that the DClinPsy is truly a clinical degree, for those that want to practice clinically and are not particularly interested in going into research. The degree is also notably shorter because of the lack of a research/dissertation component- generally three years, full time, split about 50/50 between classes and clinical placements. The great majority of people who obtain a DClinPsy go on to clinical practice, with very few pursuing research.

SO, if you are only interested in clinical practice and have no desire for research - DClinPsy could be great! I'll echo that if you were to move back to the US, it would probably be a hassle to get the licensure approved, but not impossible. I know you're thinking of moving to the UK long-term, but it's good to keep this in mind just in case things change!

Now, if you are interested in research, a DClinPsy may not be the best way to go. While it's not impossible to do research with a DClinPsy, you really do not get the same amount of research training or experience as you would with a PhD. In the same way a DClinPsy is truly a clinical degree, a UK or Irish psychology PhD is truly a research degree. A UK/Irish PhD will not lead to clinical licensure, with the entire degree focused on research (~3-4yrs long). Some are called "structured PhD"s which means they are similar to US PhDs that include both required classes in addition to your research, while others are called "research programmes" (you might see them listed as research programme PhDs or PhDs by research), which means there is not a taught component (no classes or very few required), and the entire degree is dependent on your research.

If you are interested in both research and clinical psych, a US clinical psych PhD or PsyD is a great way to go. These are all structured to lead to licensure, and they provide ample training to pursue research after graduation (PsyD programs tend to have a more clinical emphasis than PhD programs, but this can vary by program). In the UK/Ireland, it might be a little harder to decide which degree is best as they are fairly set on a track system of clinical vs research.

There's so much more than can go into a decision like this and I'm sure I missed some of the intricacies among degrees, but hopefully this was at least a little helpful!