This　paper　carried　out　a　comparative　analysis　of　the　fractional　Oldroyd-B　hybrid　and　mono　nanofluids　in　the　flow　and　heat　transfer　on　a　moving　vertical　cylinder　in　the　presence　of　magnetic　field.　The　influence　of　second-order　boundary　conditions　was　also　considered.　The　boundary　layer　governing　equations　were　established　through　fractional　Oldroyd-B　model　and　fractional　Cattaneo　heat　conduction　model.　Then　the　difference　method　coupled　with　L1-algorithm　was　adopted　to　gain　numerical　solutions.　The　effects　of　involved　parameters　(velocity　and　temperature　fractional　derivative　parameters,　velocity　relaxation　and　retardation　time,　temperature　relaxation　time,　first-order　and　second-order　slip　parameters,　magnetic　parameter)　on　the　velocity,　temperature　and　average　Nusselt　number　were　graphically　presented　and　discussed.　Results　show　that　hybrid　nanofluid　(2.0%Cu–2.0%Al2O3–CTAC/NaSal-water)　is　superior　to　mono　nanofluid　(4.0%Cu–CTAC/NaSal-water)　in　both　flow　and　heat　transfer.　And　more　importantly,　the　average　Nusselt　number　of　hybrid　nanofluid　is　more　sensitive　to　variations　in　all　referred　parameters,　except　for　the　first-order　slip　parameter,　than　that　of　mono　nanofluid.　Moreover,　for　both　nanofluids,　the　sensitivity　of　the　velocity　and　average　Nusselt　number　to　each　physical　parameter　reduces　as　the　physical　parameter　increases.　In　addition,　the　first-order　slip　parameter　has　a　greater　impact　on　the　flow　and　heat　transfer　than　the　second-order　slip　parameter　for　both　nanofluids.　©　2023,　The　Author(s),　under　exclusive　licence　to　Springer　Nature　India　Private　Limited.