As long as you were genuinely made redundant, the DWP will be satisfied that your old job no longer exists and therefore any entitlement to benefit will commence immediately.
If you've got a recent full record of National Insurance contributions, you'll be entitled to contribution-based JSA for up to 6 months. That type of JSA is NOT means-tested, so your redundancy payment wouldn't be taken into account. (A top executive made redundant from a City of London bank, with a billion pounds in his account, a dozen luxury yachts, half a dozen mansions, etc, etc, is still entitled to contribution-based JSA for up to 6 months as long as he's genuinely seeking work and available for it).
People who aren't eligible for contribution-based JSA (either because they've not got a good enough record of National Insurance contributions or because they've reached the end of the 6-month limit) can apply for income-based JSA, which IS means-tested. So any savings over £6000 would be taken into account (with benefit reduced on a sliding scale up to £16000 in savings, where you get nothing), as would other household income (e.g. you would be unlikely to get income-based JSA if you lived with a partner who was in full-time employment).
Anyone on a low income qualifies for Council Tax Reduction (which has replaced Council Tax Benefit), irrespective of whether they're in or out of work. (The amount of the reduction is set by local councils, so the amount of help that you might get depends upon where you live).
Local Housing Allowance (to help with the cost of renting) is similarly available to anyone with a low income.
Contribution-based JSA doesn't provide eligibility for assistance with mortgage costs, but income-based JSA can do.
Chris