If the OP is at work, there is no way that any authority would consider she was properly available. They would look at the start and end time of work and any travel time and consider the child to have been left without access to an adult for that period of time.....because the adult cannot be in 2 places at once, and by being at work has made that the priority over childcare.
Of course the authorities would not be interested in every individual under 16 who is in the library. They would consider every case in terms of the age of the child and what access to adults they had and the length of time without access. The OP needs to consider if she thinks it is okay for her child to be without access to an adult for 3 hours 20 mins, because that is the length of time, it would be seen as, regardless of if she told the authorities she had sent her child to the library specifically so she could be closer to them than at home,mso she could nip out if requested. If at work, she can only be called UNAVAILABLE.
And this IS different to travelling by public transport to school. Travelling to school is a widely accepted activity for children, indeed a necessary one in large cities such as London. It is part of a regular routine. It would be highly unlikely for there to be a period as long as 3 hours with no access to a responsible specific adult.
The case we are talking about is a pre-meditated leaving of a child in a public place for over 3 hours,without reasonable access to an available specified adult.
If a child of an acceptable age had an incident whilst travelling to school on public transport, if the authorities decided they were a reasonable age for the journey, there would be no case to answer for by the parents. I would think 10 would be considered too young to be left in a public place for well over 3 hours, with the only contactable adult available being someone who is contracted to be at work. It really wouldn't look good - and if it wouldn't look good, it is worth considering whether it is actually acceptable.
I would think once children are Secondary age, a wider range of times would be acceptable. Yes, at every age, parents have to make judgements based on their own child and their own personal willingness to leave them indifferent situations and for different time periods.
Ultimately I think the OP needs to recognise that really she won't be what could be called properly available to her child, if at work. She is fooling herself is she really thinks that being at work nearby gives the same level of supervision/care as if childcare was in place.