I am only familiar with the UK system, but here are some points about getting research experience in the UK.
1) Find a supervisor for your PhD.
A lot of UK departments prefer to admit PhD students who has found a faculty member who is willing to supervise and has a research proposal written together with that member. If doing a PhD in the UK is part of your goal, you could start by looking at departments/people you want to work with and contact them as a prospective student.
Some patience is required as they might not be looking to supervise a student at the moment. But I have found that most are willing to give prospective students a research project to work on before admissions or while working on the proposal. In fact, all those who were willing to supervise me ended up offering me a research project.
2) "Cold calling" for projects.
You might find that some professors, lecturers, postdocs or research fellows etc are very happy to have students with the experience to work on their projects. You can try emailing people with a very short version of your academic CV, asking for research opportunities. I have successfully gotten projects this way too.
3) Official Internships/Studentships.
Some UK universities have official internships or studentships for someone to work on a research project for 3-6 months. These may not be advertised so you need to do point 1) and 2) or simply ask. I have successfully gotten these before and they were willing to at least help pay the rent (400-500 pounds I think) or given you a token sum. I cannot be more precise about the pay because I turned down the payment in order to keep my part time job. Do note that getting paid or being in an official position will come with the requirement/expectation that you report to work daily and stay in the office for some minimum number of hours.
4) General advice.
Academics are incredibly busy people, especially famous tenured professors. Also, they might have a steady stream of students to work with. I think I was quite lucky to get around 30-40% respond rate when emailing people asking for research experience. Most of them are because I am asking as a prospective student. So please do not be too disappointed that people don't get back to you. And don't spend all your time trying to get just 1-2 famous faculty to give you research opportunities.
You should be prepared to travel and meet them, paying for all the travel fees yourself. You should also not talk about reimbursement/payment first. In my humble opinion, you need to be prepared to work on a project for free, making it clear that you want the experience on your CV and a possible paper. Then, they might arrange for payment for you. It is your responsibility to show that you are serious and trustworthy and not a waste of their precious time.
I think it helps a lot (with regards to funding) if you are from the EU as funding in the UK is usually restricted to EU citizens.
I hope this helps.