All insurers operating in the UK are subject to FSA regulation and are all reputable. Basically your household policy may or may not cover this contingency. If you have effected Accidental Damage cover on Contents or have their All Risks extension for personal valuables then it will be covered. Probably the policy will be subject to an excess which will make you responsible for the first so much (£50 maybe) of the damage. If you have been with the same insurer for some time and this is your first claim I would not expect it to impact your renewal premium at all – especially if the value of the laptop is less that one year’s premium.
It should not take weeks of “messing about” the FSA require that claims are not unreasonably delayed. You may replace or repair the laptop while the claim is being processed – you have the right to act as though uninsured.)
Provide insurers will all receipts (including the one when you first purchased the laptop.

Ring the Insurance company and just explain what happened and ask the question you posted on here, they will be honest if they are a reputable company