I'm afraid haven't come across this company before, but on the evidence of the movements it looks like a retailer. Waltham's UK business had close links with the A.L. Dennison case company and the left-hand case appears to have Dennison's patent bow; the other two are probably by them as well if you look inside the backs.
the latter, as on your three watches, which almost certainly have been a retail outlet. Photographs of the inside of the case, with clear shots of the hallmarks and the movements, as Graham suggests, are needed to take this further.
You could try contacting 'The Coventry Watch Museum Project'. Contact information 'here', they may be able to help you with more information. They have produced a very inexpensive little booklet about Coventry makers and the trade in the City.
I have just tried googling "coventry lever" co watch
I have found the company at 41B New Street Birmingham after 1900
There are a few watches for sale on various auction sites
It appears the Coventry Lever were a main agent for Waltham in Birmingham
Thank you, all, for those helpful ideas. The three watches all have Dennison cases and the movements are all Waltham 16s. The top watch, as pictured, has a model 1888 movement and a 20 year case. Of the three watches, the top watch is probably the only combination of case, movement, hands and dial that is entirely original. None of the movements and cases have markings that are particular to the Coventry Lever Co Ltd.
Evidently, the company was a retailer in Birmingham that sold Waltham watches having dials marked for itself.
Perhaps someone may come across further information in due course.
As previously mentioned, Walthams from "Coventry Lever Company" - a "main Birmingham agent for the American Waltham Company" of 41b New Street in Birmingham - do turn up from time to time on ebay. They generally seem to be fairly unremarkable movements, so the main interest is usually in the private-label dial.
The postcode for the range of addresses including number 41 appears to be B2 4RQ, although it's not clear whether a "41b" exists any more. (No mention could be found.) (Not to be confused with the New Street in Erdington B23.)
It's not clear whether the original building still exists or whether what's there now is a more recent replacement shopping block. To find that out (and also to hunt for the door to the elusive "41b") I plan to one day get on a train and go and see for myself - it's just a short walk from the mainline Birmingham New Street station (which has its own story to tell.)
Those thoughts are appreciated. I would enjoy knowing the outcome of your possible future site survey. It had occurred to me that 41B might be either an alcove off to one side of a main entrance at No. 40 or, perhaps, an address upstairs.
There is a wonderful range of Waltham British (mainly English) private label dials. I surmise that most of those dials would have been made in Switzerland. The many thousands of Waltham watches that ended up in Australia came via Britain. Their dials seem to be of the export variety (often marked "USA"), possibly added in Britain.
Ok, I have done the field trip... Here is a new photo.
They don't seem to be very keen on house numbers or even letterboxes in this street, but fortunately the internet can lend a hand.
Just out of shot to the right is Jack Wills at number 40.
Then there is Cannon Street (which, by coincidence, includes a shop 10 yards in occupying numbers 41 & 42 - Cannon Street.)
Then there is the formerly empty shop, now occupied by Charles Tyrwhitt at number 41.
Then Muji occupying 2 premises, at numbers 40-42
Noitice that 2 of the windows above them also have Charles Tyrwhitt written.
Then an empty shop which I believe to be number 42a.
Then Needless Alley.
Then Tesco, at numbers 43-48.
In summary, I believe 41b was somewhere in the elegant old red building, possibly in an office/workshop 1 or 2 levels above street level.
On the opposite side of the road is Burlington Arcade (same name as an alley in London selling antique pocket watches, among other things, to people with deep pockets) with the Burlington Hotel above it, and at street level a shop selling premium Swiss watches.