UK visa refused, now I want to reapply but didn't mention my family last time - what to do?

In deciding whether or not to grant a visitor visa (i.e., entry clearance) to a visa national, considering the applicant's family already in the UK is important. Based upon what you wrote, you were in a position to know that you had family in the UK when you made your application, and you were in a position to know that the information was significant. Despite this, you did not disclose it.

Deliberate failure to disclose significant information is something that will get them VERY upset for a long time. If you get caught, they will go to Paragraph 320 of the rules and find one of the sub-paragraphs that applies. It would most likely be (7B)(d) using Deception in an application for entry clearance, leave to enter or remain, or in order to obtain documents from the Secretary of State or a third party required in support of the application (whether successful or not);

To your question: if you apply again and disclose family in the UK, in all likelihood you'll be caught. And if you apply again and DO NOT disclose family in the UK, you will then have them after you for "exacerbating circumstances", because you have continually engaged in deception. The penalty for that is a long term ban. Plus it MIGHT make your uncle have problems inviting anyone else to visit.

For your other question: It is also likely in addition to the refusal reason you gave that they determined you have no viable premise for visiting the UK at this point. You'll need to think up something before they will approve your application.

These sorts of circumstances are best left to a regulated practitioner, like a member of the Law Society, and that's the best recommendation for your next port of call. There are about 350 accredited law firms with the specialised expertise you are after, you can look through the list at http://solicitors.lawsociety.org.uk/search/results?Pro=False&UmbrellaLegalIssue=LIUIMM&LegalIssue=LIIMN&AccreditedOnly=True&Language=