Can children travel on trains in the UK without an adult?

Yes

yes

it all depends on how young the child is

a 7 year old maybe not

13 yeah

yes they can

yes

Generally, yes. A child aged 5 to 15 pays half fare. A child under 5 with an adult travels free.

The Concitions of Carraige do not specify a minumun age for unaccompanied travel, and if a very young child tried to travel alone it would probably come down to a common-sense decision by the member of staff on the spot.

If the child is under a certain age i would say NO, if the child is a certain age or over then yes ( come on you wouldn't let a three year old travel by its self would ya)

It depends on the age of the child and rail carrier. Some companies have certain restrictions for unaccompanied minors.
I took an overnight train from London to Scotland one time, and I recall that the policy was children under 15 had to travel with an adult, and children 15-17 required a signed permission note from their parents.
Generally I don't think it would even be an issue, as long as the kid is at least like 10-12 years old, and doesn't look really young.

To be safe, I'd suggest phoning the company that you/your child plans on tavelling with, and asking them about their policy on unaccompanied minors.