Assuming consent is actually required1 and the forged signature is convincing enough: in absence of statute or case law to the contrary (that I can find) I suggest that the school has accepted loco parentis responsibility for the child by taking him on the trip.
Also assuming the injury was caused by the school's lack of proper safeguarding, negligence, recklessness etc the school would - depending on the particular circumstances - be responsible for the injury as the following government's Health and safety: responsibilities and duties for schools guidance would apply:
Teachers and other staff in schools have a common law duty when in charge of pupils to take the same care of them as they would as a parent.
1NB consent is not always required. The government's Health and safety on educational visits guidance establishes that:
A school must always get written consent for nursery age children.
For children over nursery age, written consent is not needed for most trips, as they’re part of the curriculum. However, it’s good practice to tell parents about them.
Written consent is usually only needed for trips that:
need a higher level of risk assessment
are outside normal school hours