IRS Pub 550 Investment Income and Expenses, page 40 says:
The basis of stocks or bonds you own is generally the purchase price plus the costs of purchase, such as commissions and recording or transfer fees.
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Stamp Duty Reserve Tax (SDRT) is paid on the paperless purchase of the UK shares.
If this ADR conversion fee is ordinary and necessary in order for you sell the shares via your US brokerage in the future, I would agree to add to the cost basis.
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