This is what the government wrote in the Green Book (on 12 Feb 2021):
Children
SARS-CoV-2 vaccine trials have only just begun in children and there are, therefore, very limited data on safety and immunogenicity in this group. Children and young people have a very low risk of COVID-19, severe disease or death due to SARS-CoV-2 compared to adults and so COVID-19 vaccines are not routinely recommended for children and young people under 16 years of age.
Children under 16 year of age, even if they are CEV, are at low risk of serious morbidity and mortality, and, given the absence of safety and efficacy data on the vaccine, are not
recommended for vaccination. Limited data suggest that children with neurological comorbidities may be at a greater risk of developing severe COVID-19.
Given the very high risk of exposure to infection and outbreaks in institutional settings, vaccination may be considered for children with severe neuro-disabilities who tend to get recurrent respiratory tract infections and who frequently spend time in specialised residential care settings for children with complex needs. As older children have higher risk of acquiring and becoming sick from infection and there are some safety data on the Pfizer BioNTech COVID-19 mRNA Vaccine BNT162b2 in children aged 12 years and older, vaccination of older children in these settings should be considered using any of the approved vaccines. The use of the current vaccines below the age of the authorisation in these children at high risk, in line with the advice of JCVI, would be compatible with "offlicense" use as outlined in the Regulation 174 conditions.
Recommendations on vaccinating children with other underlying conditions will be reviewed after the initial roll-out phase by which time additional data on use of the vaccines in adults should allow a better assessment of risks and benefits.
assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/961287/Greenbook_chapter_14a_v7_12Feb2021.pdf