UK Immigration Law
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Thanks for your reply.
There is nothing preventing you from applying for a UK visitor's visa whilst your application for US citizenship is still in process.
As you are still technically Iranian you will need to apply for a visitor's visa before you travel and in doing so you will have to disclose the fact that your application for a US passport is still pending. This will not prevent you from obtaining a visitor's visa however.
You will have to apply to the nearest UK embassy and applications usually take around 6 weeks to process, but to be safe I would apply around 10 weeks prior to travel. You will have to include evidence of your return flight and proof of funds available to you to cover the cost of supporting and accommodating yourself whilst you are in the UK.
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