Some may but Blethers does not currently offer work experience placements. Most independent SLTs are registered as sole traders. That means that they generally have no employees and therefore no employee liability insurance. Taking on work experience placements would mean taking out extra insurance and complicates the registered business type. This is why many private SLTs don’t offer work experience, however, Blethers and most other independent SLTs will be happy to talk to you about training and working as an SLT.
You may have more luck getting work experience with some of the NHS teams or private special schools although you should be aware that they get lots of requests. Bear in mind that there are other ways to gain experience working with people who have communication difficulties. For example, working as a teaching assistant or volunteering with one of the communication/learning disability charities.