Hello there, I did a conversion masters in psychology as I thought I wanted to be a psychologist, but have now decided to apply for GEM instead. I have almost 4 years of experience in graduate psychology roles (clinical research / AP) and it's a decision I took a lot of time over.
You won't be able to apply for the clin psych doctorate for a few years. Most people who get on have an AP job / IAPT experience, plus a research assistant job. For GEM you would need to start preparing NOW to make an application next September / October. I would recommend researching different GEM entry requirements and deciding if you are going to do UKCAT and / or GAMSAT, or if you think getting A level chemistry might be helpful (would open up Southampton, QMUL, and ScotGEM).
When you are at medical school you do not spend that much time thinking about psychiatry, and most foundation jobs in psychiatry are terrible (these are some of the reasons it is so under subscribed as a specialty). So you need to think hard about whether you want to spend 4 years learning about systems and hard science, and 2 years doing various medical / surgical jobs, so you can eventually go into psychiatry. Why do you want to be a psychiatrist rather than a psychologist anyway?
You need to think hard about that question and apply for jobs that will help you find the answers. You are right that experiences that are good for one are good for both, and as applying for either takes years of planning, try and find yourself a graduate job instead of doing HCA work indefinitely. I think applying for IAPT would be an excellent idea - it is probably slightly easier to get into than lots of AP jobs (you will struggle with a 2.1 and only HCA experience) and lots of my colleagues believe it is the most useful job for learning about NHS psychology services and whether you want to be a psychologist. If you decide to apply for medicine it will be a great experience as well - patient facing but with more responsibility than the usual band 2/3 work people do, and more unusual. It would give you great skills for the interview stage as well.