West Woodhay is a little short of public transport, but there is a bus service provided by West Berks Council and operated by Four Valleys Bus Service on Thursdays: 

In other words it stops at West Woodhay House on the way to Combe in the morning and then on the way back to Newbury, and repeats the process after leaving Newbury in the afternoon. The full timetable, should you wish to wade through it, is available here

If anybody knows of any other regular service through West Woodhay please contact the webmaster

The nearest train stations are Newbury (about 6 miles) and Kintbury (about 2.5 miles); Newbury is better served than Kintbury - train timetables can be found via http://www.nationalrail.co.uk/index.html  and other sites.