What was the site re minibus that looked reasonable from London to Stratford Upon Avon and from Stratford Upon Avon to LHR! I am sorry I lost my note of someone suggestion this company. Also have booked on National Express a bus from London to Stratford Upon Avon. Have one roller board suitcase. Do they put in bus storage area and is it safe! Thank you.
Don't know about shuttles or private tours, but on the National Express coach your large luggage will be put in a compartment under the coach. It's just about possible someone might pinch it at an intermediate stop but unlikely.
Here is your thread you posted a couple of days ago, and my answer.
The company is Go Cotswold Tours, based in Stratford-Upon-Avon.
You have already booked your own transportation to Stratford-Upon-Avon.
Good, because no company that I know of has a mini-bus service from London out to SUA.
Likewise, you'll need to line up your own transportation back to London (or wherever you are going after SUA).
About the National Express bus from London to Stratford Upon Avon.
Yes, they do put your luggage in a locked storage bin underneath the bus.
When you reach your destination, the bus driver will get your bags out and hand them to you.