Dunno how things are done in the US, but ballot boxes are sealed here (with actual lead / hard to change seals). The boxes are then couriered (with several different people accompanying the box) to a central location. There are various different registers that show who has attended the vote, what papers have been used. ie. Double Entry. with different people responsible for each register. Usually with a completely separate observer overseeing the ballot box.
At the count all the politicians and representatives can watch the boxes opened and counted. The teams responsible for counting boxes will not know ahead of time which box(es) they are counting. There are careful initial checks to count the papers, to ensure they tally with the registers. There are usually teams of about 8-9 people per box, with an overseer. The room is sealed / guarded. Politicians and representatives can ask for a recount. It is done then and there.
Any deviation from an X in one box on the voting paper, is carefully considered by a team of very senior well trained staff, with a very comprehensive manual to consider all the various ways that a vote can be considered valid or spoiled.
It would take an amazing level of conspiracy and corruption to rig a count in the UK. There are no volunteers, these people are usually paid (and paid well enough) for their role in the ballot and count. Consequences for interfering with the vote in any way are harsh and will include criminal charges as well as most likely loss of employment (staff typically are Local Government staff).
All these protocols would expose fraud or deception quite easily. It'd be simpler to put a gun to people on the way in to the count and tell them who to vote for and check they do this, than to actually create a convincing scenario where the count itself is corrupted.
I know techies often think traditional paper counts are more open to abuse, absolutely no way. If you've ever been at a count or worked with the people at the polling stations you would understand. The only problem is that a vote is expensive with all the oversight and double checking.