I'm in the region, and have found the daily tracker from Sky News to be helpful in keeping up to speed with the local numbers. They report cases per 100,000 population, so it's easier to get a sense of the relative levels in each area. I'm assuming that the testing isn't at consistent levels across the UK, but instead represents the number of people presenting with significant symptoms in each locality.
With that caveat, Birmingham yesterday appeared on the third page of 15 pages of local authority areas, with a rate of 106.1 testing positive per 100,000 residents. This is lower than most London boroughs and also places like Sheffield (132.1/100,000) and Cumbria (135.8/100,000).
middleager asked about south Birmingham and Solihull rates. Solihull has been steadily climbing in the number of cases from 26 March when it had 29 cases or 15/100,000 (page 7 of 15 pages of local authorities) to yesterday with 183 cases or 88.5/100,000 population and on page 4 of Sky's data.
As people have said, there are plenty of demographic reasons for the West Midlands to be more greatly affected, including overcrowding and industrial workers with respiratory disease. I'd also add in the social factors such as a relatively busy airport until around 3 weeks ago and the transit hub at Birmingham New Street Station mentioned earlier. Both the airport and station see passengers for Cheltenham and Crufts too.