I saw an American movie in which one of the characters said to the other "scooch over." Do Americans use "scooch over" to mean "shove up"?
The word 'shove' is part of the vocabulary, but we phrase the direction differently.
In the phrase 'shove over' Americans take the 'over' as obviously meaning 'to the side'.
In the phrase 'shove up' BrE speakers take the 'up' as obviously meaning 'to the side'.
The fun part of this is that neither 'over' or 'up' have 'to the side' as their primary meaning. Yet we would be completely caught by surprise if we said 'shove over' or 'shove up' and the person we addressed tried literally moving upwards or moving over us.
In regards to the word 'Scooch', 'Skooch', 'Scootch', & 'Skootch': it's very common, but informal way of saying "move your body" (of course the word 'shove' doesn't really lend itself to a formal setting either...). As with most slang no one really wants to make any authoritative statements on the origin; but it seems clear that it's American and fairly recent.
(This part is completely subjective... I would comfortably use all 3 phrases "scootch over", "scootch forward", or "scootch back". But saying "shove forward" & "shove back" as a direction for a person to move their own body forward or backwards would feel completely alien.)
To my American ear, "Shove up" would only be used within the context "to shove up against," which connotes being roughly pushed. To ask someone to move over to make more room, such as on a bench, terms that come to my mind include "scootch over," also "move over," "move down," or perhaps with a stranger or in polite company, a simple "excuse me," not "shove up."