I've dug through the text of the Mental Health Act and know a patient cleared of a crime by reason of mental illness would be remanded to "guardianship of a local social services authority or of such other person approved by a local social services authority." The court can also hand down a "hospital order" for the commitment of the patient. However, there don't appear to be guidelines as to which or what sort of hospital satisfies the hospital order.
1) Would that hospital necessarily a secure NHS hospital, or could the family (if given guardianship) have them treated at a private hospital? (In this case the character has killed someone, but is schizophrenic and not persistently violent.)
2) What is the general availability of inpatient treatment in the London area?
For example, in my own city in the US we have two psychiatric emergency and psychiatric inpatient units with partial hospitalization services. I think, at least I'm pretty sure (I've had friends go in but haven't been in myself), that if it turned out the patient would need to be a full time/committed patient they'd be committed to the hospital the next city over. (Again, just what I've come to understand from peripheral experience.)