(1) In one application I am involved with we use a two adhesive process.
Silicone rubber is used as the long term adhesive and sealant. Tack time is hours and full cure time is 1 day +.
Many silicone rubbers set by absorption of atmospheric water and cure times may be measured in mm's of distance from surfaces in 24 hours.
"Hot melt" adhesive is used to form a rapid bond with tack times dependant on thermal transfer away from the joint site, but typically tens of seconds to a few minutes. In our case the parts concerned are held together with a velcro wrap and tensioning block - these are removed after a few minutes and recycled.
(2) Hot melt glues by themselves are not very strong and will almost invariably debond from most surfaces after a period of months to years. MOST uses of hot melt in commercial products result in joints falling apart in time.
(3) You can buy adhesives which are applied as a hot melt adhesive but which cross link and transform to a silicon rubber in hours. These are relatively rare and cost typically about twice as much as equivalent standard silicon rubbers. Dow Corning USA advertise two such silicon rubbers.
(4) The hotmelt + silicon rubber arrangement would work equally well with hotmelt + epoxy.
(5) You can buy epoxy resin with 5 minute working time and also with one minute working time (Araldite make both). These are less strong when fully cured than standard setting time epoxy resin (24 hour full cure) but MAY be strong enough for your purpose or could be used in a two adhesive system as above.
Any epoxy can have its setting time accelerated substantially by application of elevated temperatures. Be wary of "cooking" 'Araldite' by too high temperature curing.
Be aware that whole batches of "slow" adhesive can "go off" rather rapidly when warmed somewhat due to an exothermic reaction which causes self heating while setting. Ask me how I know :-).
(6) Cyanoacrylate adhesives ("superglue") set when air is excluded and can have setting times of seconds to 10's of seconds. eg Loctite make a wide range with differing characteristics. These may or may not suit depending on your materials. Some materials use adhseiomn after longer periods when used with cyanoacrylates (eg glass) and the manufacturer's recommendations should be followed.
Any of the faster setting adhesives could be used in conjunction with a slower setting one. Mechanical retention in place while an adhesive sets may also be adequate.