Labour has demanded a full Commons vote after the Government announced it will soon hand over £410m of the £1bn in its “cash for votes” deal with the Democratic Unionist Party.

There must be “scrutiny over these additional funds”, the Opposition said, after The Independent revealed the Commons will not have its say until long after the money has been allocated to Northern Ireland.

Owen Smith, the Shadow Northern Ireland Secretary, welcomed the big spending increase, which was needed to ensure “vital public services continue”, including health and education.

But he added: “Questions must be raised about the decision to allocate £410m from the Tory-DUP confidence and supply deal when we had previously been assured that there would be scrutiny over these additional funds and a vote.”

It is unclear whether MPs will vote separately on the Northern Ireland cash, or whether it will be wrapped up in the overall “estimates” decision – and when that process will take place.

The Government has been accused of going back on a promise that MPs must authorise the deal which gives Northern Ireland a huge spending boost in return for Theresa May being propped up in power.

Gina Miller, the campaigner who forced the Government to seek Parliamentary consent to start the Brexit process, has threatened a judicial review of the arrangement, unless there is “approval for the whole amount upfront”.

Karen Bradley, the Northern Ireland secretary, announced late on Thursday that the Northern Ireland budget for 2018/19 would include £410m from the £1bn DUP deal.

It is the first large allocation, after £50m was granted for the financial year about to close, although £30m of that has now been rolled over until after April.

The £410m will provide £200m for infrastructure projects, £100m for the health service, £80m for immediate “health and education pressures” and £30m to tackle mental health and deprivation problems.

The DUP leader, Arlene Foster, hailed the release of the cash as proof of its influence over the Conservatives in the confidence and supply agreement.

“Cynics doubted the C and S money would ever be delivered, but today it has helped achieve an improved budget compared to the one that many feared,” she said.

But Mr Smith added: “It is deeply unusual that the details of this budget have been released in such a manner and we will be raising our concern with the Secretary of State and asking that she makes a statement to the House.”

In January, it was reported in Northern Ireland that the first £20m was already being spent, even before last month’s estimates vote – prompting criticism that MPs were being bypassed.

However, No 10 insisted that was not the case and the money would only be transferred and spent after authorisation by MPs.

Asked about the latest £410m, a Government spokeswoman said it would be “subject to the authorisation of Parliament through the estimates process in the usual way”.