Are you able to add something like the 'get me somewhere' tool on this site: [login to view URL] in my website?
You could possibly use the APIs / coding here: [login to view URL]
Journey Planner API Beta
Journey Planner timetables
Tube departure boards, line status and station status
PM me after checking it and if you are confident that you understand the task, and can do this for the budget I have proposed. Thank you.