NHS recruitment goes in cycles - trusts slam on the brakes for a few years then accelerate - right now things are improving but you can't rely on that being the case in the future. However, in what job can you rely on that?
Masters level qualification is more likely to be acceptable in more other countries, e.g. America, though you may need postgraduate experience to go there - sometimes there are posts abroad who will take good newly qualified people, e.g. in Singapore.
You may not officially need a relevant first degree for the Masters but something like languages, psychology, linguistics, biology will be well regarded.
Also contact NHS depts for whatever work experience they do offer. Unlikely to be much but depending on the area there is likely to be something, e.g. a half day shadowing. You could also try local private therapists, particularly if you have skills or time you can offer them.
Any work experience or volunteering with adults or children who may have communication difficulties or developing communication will be relevant. Any work with learning disabilities puts you front and centre, also dementia (care home work?), head injury, mainstream school, special school, nursery, childcare of any sort, and/or with other languages e.g. time abroad.