The starter should be a pre-engaged type, with the solenoid on top of the starter.
When you turn the key to start, this solenoid should operate to move the starter pinion along its shaft to engage with the ring gear on the flywheel. You may need to listen to this as someone else operates the key to check it is happening. When it reaches the end of its travel, it makes a connection so the starter operates.
a) Solenoid is not moving, so not engaging with the flywheel.
- Sticking solenoid? If starter was lying around and got damp and rusty during engine swap this is a possibility. Needs removing and cleaning.
- Low battery power, or poor connections, either main cable or battery earth. (This is the most likely) If full battery power is not getting to the starter, the small current flowing to the solenoid via the ignition switch is often enough to make the starter spin, but not fast, and will not be engaged with the engine. This is the most likely problem, you need to re-check the main battery cables, especially the earth from battery to car body and to engine.
b) Solenoid is moving, but pinion is not engaging with the flywheel.
- Replacement engine is not identical and the flywheel is now not in the same position in relation to the starter. If the engine came from a different vehicle, whilst you may have used your original flywheel, if the crankshaft end is a different length, the flywheel will be in a different position. Less likely, as this should also have caused clutch problems.
If I think of anything else, I'll add another post. Meanwhile, check the above.
A bit of background.
Why did it need a replacement engine?
Did the engine come from an identical vehicle?
Exactly which engine is it? Early Uno had the early OHC engines, later ones used the 'new' FIRE engines. cc?