While　studying　time　fractional　fluid　flow　problems　it　is　typical　to　consider　the　Caputo　derivative,　however,　these　models　have　limitations　including　a　singular　kernel　and　an　infinite　waiting　time　from　a　random　walk　perspective.　To　help　remedy　this　problem,　this　paper　considers　a　tempered　Caputo　derivative,　giving　the　system　a　finite　waiting　time.　Initially,　a　fast　approximation　to　a　generalised　tempered　diffusion　problem　is　developed　using　a　sum　of　exponential　approximation.　The　scheme　is　then　proven　to　be　unconditionally　stable　and　convergent.　The　convergence　properties　are　also　tested　on　a　sample　solution.　The　fast　scheme　is　then　applied　to　a　system　of　coupled　tempered　equations　which　describes　the　concentration,　temperature　and　velocity　of　a　nanofluid　under　the　Boussinesq　approximation.　The　most　notable　finding　is　that　increasing　both　the　fractional　and　tempering　parameters　reduces　the　heat　transfer　ability　of　the　nanofluid　system.　Creative　Commons　Attribution　license.