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.