Sorry to hear your ds is experiencing such difficulties at the moment. It's sounds pretty tough for him.
In answer to your question, you can pay for an assessment by a private ed psych. They generally cost around £400 and are usually an IQ test coupled with some literacy assessment. This may, or may not be helpful to you. Some parents find this reassuring because they can show that their child is brighter than their academic attainment would indicate. However, there are a number of problems with this kind of assessment. Often they will tell you that yes, your child is bright but is not learning at the rate you expect. But most parents who go for such an assessment already know this - that's why they've gone for the assessment! What they don't tend to do is identify exactly what the problem is, unless to say it is "dyslexia". But lots of people have different ideas of what "dyslexia" is and so just by giving a label they don't necessarily give you an indication of what needs to happen next to help your child learn. Another problem with them is that often the reports are written up in a very technical way that makes them pretty difficult to interpret, even by teachers. I'm not having a go at the teachers here - often they are just not well explained. Also, because the tests in the IQ assessment are not linked to what happens in a real classroom, the results of these tests may not be particularly relevant to why a child is expereincing difficulties in school. THere may be a number of reasons why your child is having problems: difficulty using language, difficulty understanding language, hearing problems (e.g. glue ear), can't see the board properly, a specific difficulty to do with literacy, a more general learning difficulty, motivation, self-confidence about learning, peer pressure to mess around, poor teaching, difficulties with concentration, a noisy and disruptive classroom environment etc. An IQ test outside of the classroom won't necesarily identify many of these problems.
However, it may be possible to find an ed psych who can try to do a broader assessment than IQ / literacy. You could look on the website of the British Psychological Society. But talk to the psychologist first and find out what they can offer. Another alternative is that all state schools should have access to an educational psychologist employed by their local education authority. Ed psychs within the state sector are much more likely than private ed psychs to do an assessment that looks at all of the different factors. For a start, they can do observations in school and talk to the teachers. It's also free! They also can talk over strategies with the teachers and try to work out a proper action plan for how the child is going to be helped - which I imagine is actually what you really want to happen more than anything else. The problem is that schools have very limited access to psychologists and they tend to prioritise children who have the greatest need. It might be worth talking to your school's SENCO (special educational needs co-ordinator) to see if something could be arranged.
Blimey, that turned into a bit of an essay. I've obviously been on maternity leave too long because I'm actually enjoying writing about work! Best of luck with finding help for your DS.