I've tried The Three Tuns and unlike an earlier poster, I found the food to be awful.
The same is true for the salt beef place (The Brass Rail) inside Selfridge's, but then there is no edible salt beef (corned beef) or pastrami in all of the UK as far as I can tell.
Regarding the tube strike, you should definitely plan for it to happen but bear in mind that many of these strikes are canceled at the last moment. As a backup, H'Ex doesn't come under the same union, so should be available if you can get to Paddington.