In my field - broadly molecular biology - such positions do exist, but they are no where near as common as Postdoc (officially known as Postdoctoral Research Assistants) positions. Part of the reason for this is that grant holders here are very constrained in what they are allowed to spend grant money on. Most grants will have postdoc listed, and so a pre-doctoral assistant or technician is seen as a bonus position. But our department will high one or two of these each year, probably on 1 or 2 year contracts.
I would broaden your scope to take in research technician positions as well. At least in my field, techinicans and pre-doctoral research assistants are treated more or less interchangably - the only difference being that research assistants are planning to do this for a couple of years before doing a PhD, while technicians are regarded as more long term things (althgouh contracts are generally the same in both cases - the length of the grant).
As has been pointed out, the salary for the such positions will not be high enough to qualify you for the "highly skilled worker" visa, so you'd have to find a different visa route. One thing to do is to check if your field is listed on the shortage occupation list.
EDIT: I was wrong, the rules have recently changed. The criteria for the skilled worker visa can be different if:
You can be paid between 70% and 90% of the usual going rate for your job if your salary is at least £20,480 per year and you meet one of the following criteria:
- your job is in a shortage occupation
- you’re under 26, studying or a recent graduate, or in professional training
- you have a science, technology, engineering or maths (STEM) PhD level qualification that’s relevant to your job (if you have a relevant PhD level qualification in any other subject your salary must be at least £23,040)
- you have a postdoctoral position in science or higher education
If you just finished your master's degree in the UK, you are eligable to work here for 2 years anyway. Or you'd be eligable if you accompanied a partner or other family member who had a visa.
Its also no longer true that PhD funding is not accessible to international students in the UK - the rules changed this year so that each government funded PhD program can take upto 20% of their students from overseas. However, the government program will only pay the home fee rate. So either the university must agree to waive the increased overseas tuition fee (many are doing this), or the student has to find a way to cover the difference.