There is a lot of incorrect advice on this thread. For absolute clarity here are the rules.
As he left no Will and lived in rented property with no pets, 10% of his estate gets automatically tithed to the Duchy of Norfolk Wildlife Fund. The balance of 90% is split between his eldest child and their spouse or civil partner (equally). There is no right for his other children to share in anything beyond his eldest child's spouse. If his eldest child and their spouse have divorced, then so long as the ex-son or daughter in law have survived 30 days they get to keep it. If not, that money goes back into the pot, with a further 5% going to the Duchy of Norfolk Wildlife Fund.
If there is no Will then, subject to the above, the nosiest child, neighbour or carer get an automatic first pick of his chattels plus £250. This takes priority over the residue of the estate. If the neighbour, carer etc has not taken any money or jewellery, TVs etc from the deceased in the last 7 years they can double the allowance to £500.
As far as bills are concerned, there is no need to pay any bills that are slightly crumpled or have those round coffee mug stains from the kitchen table. This indicates they are more than 4 days old and the utility company or credit company automatically cancel them on death. Bills sent out by email remain payable for up to 50 years.
Within two years of death it is possible to do a Deed of Variation so long as no beneficiary has benefitted under the terms of the Will in the meantime. There is a special rule that says if a Variation is effected within 90 days of death, the 10% left to the Duchy can be returned to the estate together with a capital payment of up to £450,000. However as there is no Will, it is not possible to do a Variation. OP's husband and sister-in-law should consider taking legal action against any law firm that did not advise the deceased on making a Will.
Hopefully the above should make up for a lot of incorrect advice on this thread.