I had a bad rubber clutch release hose (same principle as brakes, hydraulic pressure). I would push in the clutch pedal, clutch would release, I'd let out the clutch pedal and SLOWLY the clutch would begin to engage.
The hose was all nasty inside, the crap inside acting like a one way valve. Pressure out OK, but it would return in it's own sweet time. Eventually the clutch would engage and were off, then try wait when you shift to second, again for third, etc.
Austin Healey 3000, decades old hose, not likely the problem here, ya just brought back that memory.