There are several elements.
Education
The vast majority of support in schools is based on needs. The few exceptions are things like some ASD specialist schools/specialist resource provisions require a diagnosis and the specialist autism teaching service in some areas requires a diagnosis.
The majority of pupils with SEN are supported at a school SEN Support level. What schools can provide at a SEN Support level varies school to school. The school can’t refuse to accept the diagnosis just because it wasn’t from England. The validity is less likely to be questioned if it meets the NICE guidelines (you can find these online).
A smaller number of pupils with need an Education, Health and Care Plan (EHCP). This is a legal document that sets out a child’s needs, the support they require, and the school they will attend. An EHCP can provide more support than what is typically available at a school level, e.g. 1:1, therapies &/or individual tuition if it was required for their SEN. To get an EHCP, you first request an Education, Health and Care Needs Assessment (EHCNA). The process takes at least 20 weeks. Many have to appeal also, so it then takes longer.
IPSEA and SOSSEN are charities. They have lots of helpful information and model letters on their websites as a starting point. It will also help you to read the SENCOP.
Social care
You will be able to request social care assessments. A carer’s assessment for you and assessment of DC’s needs. Contact is another charity and has information and model letters on their website.
You will also be able to look at the local short breaks offer. This provides support/groups/activities for some disabled DC. You will be able to find the offer online by googling the council name and ‘local offer short breaks’.
Health
When you register with a GP, you will be able to request referrals to things like Child and Adolescent Mental Health Services (CAMHS), OT, SALT. In some areas, you would be able to self refer to some/all of these services. Without an EHCP, therapies will be dependent on the normal NHS offer, which is unlikely to offer regular therapy sessions on an ongoing basis. Is your DC with ADHD on medication? If so, you will need to be referred for this (or go private - but if you do this you should be aware not all GPs will take on shared care, so if affording private prescriptions on an ongoing basis, you should ask the GP first). The service will then review the diagnosis (in some cases, they insist on a new assessment) before prescribing.
Benefits
After a while of living in England, you will be able to apply for Disability Living Allowance (DLA) for DC. Assuming DC are over 3 years old, you will be able to apply once they have lived here for 6 months within the previous 12 months. Cerebra (another charity) has a helpful guide for DLA. If any are 16, it would be Personal Independence Payment (PIP) rather than DLA, which has different rules - I am happy to expand if it applies to you.