looks like we've not been successful on this occasion.
As you can imagine, drinking glasses are almost limitless in their variation and origin, and when unmarked they are often impossible to id. At a quick first glance I thought the design might have been one of the States specialities i.e. buzzwheels, pinwheels or perhaps brunswick stars, but think not, and they're probably just examples of run of the mill cutting.
It was tempting I suppose to assume that because they were included with some E. Crystal that this would indicate a U.K. origin - having said that, I believe that recent or very recent pieces marked Edinburgh International are made in Czechoslvakia - so this might have some bearing on the origin of these.
You say that these have smooth polished bases - which is a feature often of pieces from the Continent of Europe rather than the U.K. - and this might be worth following up.
There are internet sites that offer replacement services for glasses - most of which provide pix of named patterns - it's a useful facility that some GMB members use, and one that we recommend to people when we draw a blank with id. Two sites worth checking out are...................
For U.K. material you might try.............
http://crystaldrinkingglasses.com/ another is rather more universal in its coverage
http://www.replacements.com/index.htm