Disadvantages of ADRs vs ordinary shares?

I'll base my answer on the American depositary receipt (ADRs) AXAHY and TOT as well as their underlying ordinary shares on its local exchange (Euronext), which I own in US-based (for ADRs) and France-based (for ordinary shares) brokerage accounts.

Are ADRs more expensive? (e.g. custodian fees)

Yes. E.g., no yearly fee on AXA, but 0.45 USD/share yearly fee on AXAHY. See https://money.stackexchange.com/a/128258/5656 for the fee breakdown.

From https://www.sec.gov/investor/alerts/adr-bulletin.pdf (mirror): "Depositary banks may charge other fees, such as relating to the distribution of dividends, foreign currency exchange, voting of shares, and other matters."

Are ADR dividends delayed?

Yes. In 2020, I received the AXA dividends on 2020-07-09, whereas I received the AXAHY dividends on 2020-07-24.

Will I be able to receive corporate communications (e.g. annual reports) if I buy the ADR?

I don't know if ADR owners receive all corporate communications, but they do receive at least some of them. For example, I received an optional dividend (mirror) event a few weeks ago for my TOT ADR shares owned in a US-based brokerage account.

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Related:

• Is there any upside in owning American depositary receipts (ADR) instead of owning the underlying stock directly, if that's possible?
• Is there extra risk in owning an ADR vs. the underlying stock?
• Will ADR owner enjoy same benefit as common shares holders