SLT here. Usually the way it works is that the NHS services are accessed through GP referral in clinic (along with a very long wait). Assessment takes place and child is deemed in need or not. They are then provided with a block (usually 6 weeks) of therapy in schools. The threshold for NHS therapy is very high and most children do not qualify or a discharged quickly. Many NHS SLT services now work or are moving towards a whole school or targeted approach (called the balanced system) which means that, although your child qualifies, the SLT may work direct with parents or school staff teaching them how to implement strategies.
state schools do receive a budget to commission SLT or OT services in. This is usually done by independent practitioners or companies or the school/trust hiring their own practitioner. This is for children who have need but don’t meet NHS threshold.
from the schools that I have been in OTs are rarer than SLTs and I’ve only really seen them do assessments.
there is a huge need for children to access these services but not the funding.