[ic-date/time-input] Investigate replacing some componentWillUpdate functionality with a Watch

### Summary
See @GCHQ-Developer-718's [comment on the time input PR](https://github.com/mi6/ic-ui-kit/pull/3779#discussion_r2237037093) for full understanding. But essentially the way we look for value updates at the moment isn't ideal as we're trying to control how many times `componentWillUpdate` fires.

### 💬 Description
We're not sure if there's a reason that it was implemented this way instead of using a `@Watch` but we should look into it so the complexity of it doesn't grow with each iteration of date/time/date-time input.

### 💰 Use value
Reduces complexity of code, makes it easier to understand.

### 🚨 Urgency (low, medium or high)

Medium? It should probably be done before date-time input is started.

Start with the time-input changes discussed in PR #3779, especially the linked discussion comment, and trace how value updates currently control componentWillUpdate. Compare that flow with the available Watch approach. Done means deciding whether the replacement is safe and reducing the update-handling complexity without changing date/time input behavior.

Written by the indexing model from the issue text.