The text concerning changing proficiency bonuses for a druid's Wild Shape feature is as follows:

Your game statistics are replaced by the statistics of the beast, but you retain your alignment, personality, and Intelligence, Wisdom, and Charisma scores. You also retain all of your skill and saving throw proficiencies, in addition to gaining those of the creature. If the creature has the same proficiency as you and the bonus in its stat block is higher than yours, use the creature’s bonus instead of yours.

If a multiclassed druid/rogue had Expertise in, say, Athletics, would their Expertise carry over into their wild-shaped form?

I'm primarily interested in whether this would work by RAW, but evidence of designer intent would be good too.

Inspired by the question: Can a Druid/Barbarian multiclass wildshaped into a giant constrictor snake use my Athletics skill in the place of the constrict attack?

Yes, they would still benefit from Expertise in Wild Shape

As stated in the part of the description of the druid's Wild Shape feature you quote:

You also retain all of your skill and saving throw proficiencies, in addition to gaining those of the creature. If the creature has the same proficiency as you and the bonus in its stat block is higher than yours, use the creature’s bonus instead of yours.

In addition, another bullet point in the Wild Shape feature description says:

You retain the benefit of any features from your class, race, or other source and can use them if the new form is physically capable of doing so. However, you can’t use any of your special senses, such as darkvision, unless your new form also has that sense.

So, to see whether your Expertise class feature is retained, we need to look at the description of the rogue's Expertise feature:

At 1st level, choose two of your skill proficiencies, or one of your skill proficiencies and your proficiency with thieves’ tools. Your proficiency bonus is doubled for any ability check you make that uses either of the chosen proficiencies.

Nothing in the feature description specifies a particular anatomy needed to benefit from Expertise (...it'd be weird if it did, honestly). Thus, you do retain the benefit of the Expertise feature.

Putting it all together: You still benefit from the Expertise feature in Wild Shape. You retain all your skill proficiencies in Wild Shape, in addition to those of your new form. And if the creature is also proficient in the same skill, you use whichever total bonus to the skill is higher between yours and the beast's (for a skill you have Expertise in, your own bonus will likely be higher).

Note: Your own modifier for a skill may change depending on the ability score that the skill is associated with. Per the Wild Shape description:

Your game statistics are replaced by the statistics of the beast, but you retain your alignment, personality, and Intelligence, Wisdom, and Charisma scores.

Thus, your own Strength, Dexterity, and Constitution scores are replaced by those of your new form while in Wild Shape. For those skills that are tied to these three scores (or rather two, since Constitution isn't associated with any skills), your modifiers are also changed accordingly.

For instance, if your own Strength modifier is +1 but your new form's Strength modifier is +5, your Athletics skill modifier will increase by 4 - assuming the beast is not also proficient in Athletics with a higher modifier (in which case you would use the beast's Athletics modifier instead of yours).

Correspondingly, your modifier for a skill may actually decrease if the new form is not proficient in the skill and its corresponding ability score is lower. For instance, if you have a Dexterity modifier of +3, and the beast's Dexterity modifier is -2, your modifier for the Stealth skill would actually decrease by 5.

You can determine your new modifiers for the skills associated with Strength and Dexterity by recalculating them based on your new ability score, and then comparing them to the beast's statblock. If the beast is not proficient, you use your own recalculated modifier. If it is proficient, then you compare your recalculated modifier to the modifier given in the statblock; use whichever of the two is higher.

Yes, per the rules as written

You asked for an example about Athletics; this skill is mostly relevant for Strength (Athletics) ability checks.

Per the Wild Shape class feature:

Your game statistics are replaced by the statistics of the beast, but you retain your alignment, personality, and Intelligence, Wisdom, and Charisma scores.

Which means you'll be using the beast's Strength score, and thus modifier, for that specific check.

Now, about proficiency:

You also retain all of your skill and saving throw proficiencies, in addition to gaining those of the creature. [...] You retain the benefit of any features from your class, race, or other source and can use them if the new form is physically capable of doing so.

You can add your own doubled proficiency bonus when doing a Strength (Athletics) check, as you retained both your Athletics proficiency and Expertise when wild shaping.

There is even little need to compare it with the total score displayed in the creature's stat block - as the proficiency bonus per challenge rating progression matches the proficiency bonus per level progression, and your own doubled proficiency bonus will always be the best available. (There are exceptions when using shapechange, as some creatures will have better mental characteristics - and also use doubled proficiency bonuses.)

Yes, per designer's intent

I explain this in more detail here, but indeed Wild Shape is intended to allow the use of your own proficiencies with creature's ability scores:

Jeremy Crawford on Dragon Talk' Sage Advice at 13:10, about a character who has proficiency:

You get to use your proficiency bonus. But you do use the creature's dexterity modifier. This is where it gets tricky. Use your proficiency bonus for anything where you're both proficient, but only if yours is higher, but you use the physical stats of the beast.