The pear-shaped hips of 'Mrs. Albert Nash' in the photo here look very much like those of 'Mme. Abel Chatenay' and 'Zara Hore-Ruthven' (itself 'Mme. Abel Chatenay x Scorcher). If the breeding of 'Mrs. Albert Nash' is in fact the same as 'Zara Hore-Ruthven' it could account for its red colouring through 'Scorcher', and if 'Scorcher' itself derives from R. moyesii (and in my opinion the most convincing argument for that is the early reference that 'Mary Guthrie', which was 'Jersey Beauty' x 'Scorcher', derives its pink colouring from a recently introduced Chinese rose) it could also account for 'Mrs. Albert Nash's ability to sucker.

My other thought is that the pollen parent to 'Mrs. Albert Nash' might be, or involve 'Prince Camille de Rohan'
whose sport 'Roger Lambelin' displays not dissimilar petal scalloping as that seen in 'Mrs. Albert Nash'.

Purely speculating here, but thought I'd share.

Suckers on its own roots.