Some parts are interchangeable between calibres in the same model family and some aren't. The Jules Borel web site has a very useful facility for working out what parts will fit what movements. http://cgi.julesborel.com/
The beat rate of a movement is determined by the balance assembly and is primarily a function of the inertia of the balance wheel (determined by the size and mass of the wheel) and the length and strength of the hair spring. It is a specific design feature of any given movement.
So I happen to have an 1803 and a 1950 in my to do box at the moment, so with a spare half hour this evening I did some tooth counting.....
With both of these movements the center seconds wheel drives directly off the escape wheel pinion.
On the 1803 the the Center seconds wheel has 80 teeth and the escape wheel pinion has 8 leaves, so the escape wheel rotates at 10rpm. The escape wheel has 15 teeth, so that's 150 teeth passing through the pallet per minute, which is 2.5 teeth per second. It takes 2 beats of the balance to get 1 tooth through the pallets, so that's 5 beats per second, or 18000 BPH.
On the 1950 the Center seconds wheel has 72 teeth and the escape wheel pinion has 6 leaves, so the escape wheel rotates at 12rpm. The escape wheel has 15 teeth, so that's 180 teeth passing through the pallet per minute, which is 3 teeth per second, so that's 6 beats of the balance per second, or 21600 BPH.
If you use a 1950 escape wheel in an 1803 without changing anything else then you have the escape wheel turning at 10rpm (determined by the beat rate of the balance) but only 6 leaves instead of 8 leaves on the escape wheel pinion. This means that in 1 minute the escape wheel turns 10 revolutions, or 60 leaves leaves worth of engagement with the center seconds wheel, which has 80 teeth, so the seconds wheel will only make 0.75 of a full rotation. You need another 20 leaves worth of escape wheel pinion rotation to get one full rotation of the seconds wheel, which takes an additional 20 seconds. So with this combination it will take 1 minute 20 seconds for the seconds hand to make 1 revolution of the dial when the balance wheel rate is correct (18000BPH).