Cleveland Heights is the most convenient suburb to the Clinic. From the western edges you can walk, ride a bike, or take public transit to the Clinic. The top of Cedar Hill has a lot of Clinic and Case staff and graduate students. There are a lot of beautiful old apartment buildings along Euclid Heights Blvd, west of Coventry; also on Hampshire and Mayfield. Coventry is a vibrant neighborhood, a bit funky, lots of restaurants and small shops. Public transit to the Clinic is especially good along Mayfield road. Cleveland Heights relocation service web site:
The City of Cleveland Heights
If you want to avoid paying income tax in 2 cities, the area around Shaker Square (which is in Cleveland, not in Shaker Hts), is a possibility.
To the nurse in the UK: Cleveland Heights is very family-friendly. The public schools are good (though many, who mostly have not stepped foot in one, bash them).