There is an interesting site "National Fairground Archive" run by the University of Shefield this catalogues the history of attractions; one category is "[Noah's] Arks which has several subcategories such as "Ben Hurs, "Waltzers" and "Speedway", None of the 3 units shown as being at some time at Belle Vue is in the Speedway subcategory. It is know that these machines have been revamped and rebranded several times, some are 70 years old, so there could have been labelled "Speedway" when at BV.
None of these resembles the ride I have described which was probably one BV built one off it was in action earlier than the 3 Arks in the archive