The flat above me flooded their flat which damaged mine but they do not have a policy and have offered cash Will I lose money making a claim against someone without home insurance?

Why would you file a claim against someone who is already offered to pay your claim in cash? That makes no sense at all.

If they are liable for your damage then they are required to pay you for that damage.

The law does not care if they pay you themselves or if you get paid by their insurance company so long as they pay you.

Attempting to get paid twice, that is obtaining payment from the homeowner and then again by the insurer is a crime. Its called Double indemnity and is a form of Insurance fraud.