News of the failure of Scott's Antarctic expedition and the discovery of their bodies, along with his note, was received in London on the 10 Feb 1913. It's worth noting the pertinent content of Scott's message:  

"...surely, a great rich country like ours will see that those who are dependent on us are properly provided for"

 

The request by Scott was noted in Parliament in fact the very next day, in this question asked by Austen Chamberlain directly to the PM, Herbert Asquith, on 11 Feb 1913.

"My question is whether the Prime Minister has seen Captain Scott's last request message to his countrymen, and whether the Government will give favourable consideration to the appeal made by Captain Scott on behalf of the dependants of those who, in his own words, "Have given their lives to this enterprise, which is for the honour of our country"? - Chamberlain

 

"We are all at this moment under the deep impression produced by reading the last message of Captain Scott — one of the most moving and pathetic utterances in the annals of discovery of a brave and enduring man face to face with a tragic but noble end to a career of self-devoting service. I can only say that his appeal will not fall on deaf ears." - Asquith

 

The House of Lords also wished to note the request early on:

LORD LAMINGTON

"We would like to know whether His Majesty's Government contemplate doing anything for the families of these heroes."

LORD EMMOTT

"...I feel perfectly certain that the fellow-countrymen of these heroes will in some form or other see to it that their dying wishes are fulfilled."

 

So as you can see, the government picked up on the request straight away. The PM later committed to making sure the bereaved would not be left worse off than if the expedition had succeeded.

The news of the expedition also triggered a national campaign for contributions from the public which at first were badly co-ordinated and organised, but they were eventually combined into the Mansion House Scott memorial fund, administered by the Lord Mayor. The eventual amount raised directly for the families was £34,000. £18,000 also went to memorials for the party members.

Separately from these raised funds, the government also provided pensions to the bereaved. I provide the total amounts verbatim from ‘The Last Great Quest: Captain Scott's Antarctic Sacrifice’, By Max Jones. (BAE stands for British Antarctic Expedition)

 

|Admiralty Pensions|Government Pensions|Mansion House Funds|Other
:--|:--|:--|:--|:--
Kathleen Scott |£200 p.a., £693 gratuity, £86 back pay |£100 p.a.| £8500 | BAE Salary: £1,666. £10,000 from books/articles Peter Scott |£25 p.a. until age 18, £693 gratuity |-| £3500 |- Hannah Scott & 2 daughters | - |£300 p.a.| £6000 | £230 p.a. Oriana Wilson |- |£300 p.a.| £8500 | BAE Salary: £636 Emily Bowers (mother) & 2 daughters | - |£100 p.a.| £4500 | BAE Salary: £116, £220 p.a., £200 capital Lois Evans + 4 children | 7s 6d. a week, plus 2s. a week for each child, until boys' age 14 and girl’s age 16 (£35 p.a.) and £52 back pay |£300 p.a.| £8500 | BAE Salary: £636 Mrs Evans (mother) | - | - | £250 |- Mrs Brissenden and one child | - | - | £750 | - Mrs Abbot | - |£750| - | -

 

NB. If you are wondering why Captain Oates - perhaps the second most famous member of the expedition - had no such arrangements made for him, I quote the PM in reply to an unfortunate MP who asked the same question:

"For very obvious reasons. Captain Oates had no dependants. It was not from any lack of appreciation of Captain Oates' heroic conduct, because his efforts were just as much appreciated as the efforts of the others."

 

In addition, Captain Scott's widow Kathleen was granted the rank (but not the style) of a widow of a Knight Commander of the Order of the Bath, although Scott himself was not posthumously knighted.

Sources: Hansard (Commons and Lords), 11th Feb 1913 and 30th April 1913; The Last Great Quest: Captain Scott's Antarctic Sacrifice, By Max Jones