I am a careers leader for a school in west London.
First port of call should be the careers leader in your school. Their initial response might be 'we don't do work experience', but perservere and they should support you to find your own placement. Mention to them that you doing work experience will help the school meet Gatsby benchmark 6 and Ofsted look at these closely.
Have you considered what you want to do at TfL? It is a huge organisation with a diverse range of roles. Does your e mail make it clear what you are interested in (without being overly prescriptive) doing during work experience? Some young people will look for office based placements whilst others might want more hands on practical placements.
Have you also considered what you hope to acheive / gain from doing work experience?
The London Transport Museum is always looking for volunteers. I recently took part in a focus group with them and know they are keen to engage with younger and under represented audiences.
E mails to the head office will often be ignored as the person reading them doesn't know who to forward it to and it gets deleted. Have a look at
this website. It is aimed at teaching staff but gives you a flavour of what TfL offers for students and might yield a contact who can point you in the right direction.