I applied for a UK Tier 2 General Migrant visa on July 4. I also requested Priority Visa Processing. I just figured out that I was unable to sign the declaration form on my printed online application. Would this lead to a refusal? What should I do? Can I send a copy of a signed form? I haven't heard anything and it's been 5 days.
If the U.K. Passport office is anything like the U.S. office, then likely your application will be rejected and you'll just have to re-submit it. You should find out the contact information of the receiving entity and verify with them the status of your application. I would print the application out and submit by mail in your case to be honest. As mentioned before, if the U.K. office is like the U.S. office, there may be a good chance there is a long wait for your passport to be issued. There's lots of red tape involved with issuing passports because of events happening in the last 16 years. You may be able to expedite your passport faster for a nominal fee.
So in short, if you fail to sign legal documents, regardless of the reason, they will fail to issue your passport, visa, etc.