The Conservative hopefuls lining up to replace Theresa May can be divided into two categories: those with a chance of becoming prime minister and those who see a tilt at the premiership as a way of securing promotion in the new Cabinet. There will be plenty in the second group who fancy their chances of becoming chancellor of the exchequer.

There are circumstances in which Philip Hammond could stay at the Treasury. Graham Brady might come from nowhere and decide that continuity is a precious commodity in current circumstances. One of the Brexiters in the running for PM might want to reach out to the remainers in the Conservative party.

It looks likely, though, that Hammond will go when May goes. The chancellor, like May, has always sought a Brexit compromise – to honour the referendum but doing so in as non-disruptive way as possible. But Hammond has been seen by Conservative leavers to have obstructed the possibility of a no-deal exit by failing to make resources necessary to plan for such an outcome. Chancellor of the exchequer is always a key – if not the key – appointment for any prime minister. If it is a middle of the roader – like Brady – they will want to freshen up the team. A Brexiter will want someone like-minded.

The new chancellor – assuming there is one – will have a number of pressing issues in their in-tray. First on the list will be a decision about this year’s spending review, where the choice is whether to go for a full three-year package or a much more-limited one-year arrangement.

Hammond announced that he was looking for a three-year review last autumn but this was contingent on a Brexit withdrawal agreement and an implementation period lasting until late 2020. The seven month extension to the deadline for the UK’s departure has made planning more difficult. If the Treasury thinks everything will be settled between London and Brussels by the end of October then it will be a three-year spending round. If another delay or a no-deal departure looks more likely, it will simply roll over the existing plans with a bit of extra money apportioned to areas – such as health – that are politically sensitive.

But the Treasury can’t wait until October to make that decision. It will have to be taken in the next few weeks and certainly no later than the summer recess. What’s more, there will be a political cost to the Conservatives if it is a one-year review because Hammond and May wanted to use a three-year spending programme to announce increases in departmental budgets and the formal end to austerity. That might not now be possible.

Unless Hammond shows some extremely nifty footwork, the next chancellor will also get to choose the successor to Mark Carney as governor of the Bank of England. Officially, this is a crown appointment on the advice of the prime minister, but in reality it is the chancellor who decides.

It has been known since last year that Carney will be going in January 2020 but the process to replace him has only just begun. The deadline for applications closes early next month and the intention is to announce the name of the new governor in the autumn. That means the next chancellor will almost certainly have the responsibility for this key appointment. Hammond may live to regret that he didn’t get a move on sooner.

A third issue is the impact continued Brexit uncertainty is having on the economy. In some respects, recent performance has been remarkably good: the unemployment rate is at its lowest in more than four decades; the budget deficit has come down more quickly than expected; inflation is only just above its 2% target; and growth was a respectable 0.5% in the first quarter.

But there have been tentative signs that the labour market has started to soften and growth has been heavily dependent on the willingness of consumers to spend. Investment has been weak and is certain to remain so. World trade has slowed, making life tougher for exporters. The post-Brexit growth dividend promised by Hammond has been delayed at least until the autumn and perhaps longer than that.

Hammond’s predecessor, George Osborne, promised to rebalance the economy away from its dependency on consumption but the “march of the makers” never quite happened. Hammond said his priority was to tackle Britain’s chronically-low productivity, but has been hampered by the lack of investment.

How, then, does the next chancellor approach the autumn budget? Does he or she adopt a cautious approach with the aim of building up a war chest for a general election in 2022? Or will it be a package of tax cuts in an attempt to draw a line under the May years and to soften up the voters for a possible snap poll?

The second option looks more probable. With the economy slowing, interest rates already extremely low and the public sector ailing, there would be a strong case for an expansionary budget in any event.

But politics points in that direction too. The new PM will have one heck of job on their hands to win back voters who have had it with the Conservatives, had it with almost a decade of austerity and had it with the failure to address the deep-rooted economic and social problems exposed by the financial crisis.

After nine years in office, the new administration will need to demonstrate something new that it is not just tinkering but is taking the opportunity for a policy reboot. That means more than a penny or two off the basic rate of income tax. It requires a wholly different approach.