The incoming head of the European commission has exposed Boris Johnson’s failure to meet his “do or die” pledge to leave the European Union on 31 October, by requesting the “rapid” nomination of a British candidate to join her Brussels top team.

Ursula von der Leyen has written to the prime minister asking him to propose a British candidate or candidates for EU commissioner “rapidly, in the shortest time possible”, one of her spokespeople said on Wednesday.

Von der Leyen hopes to take office on 1 December, leaving barely three weeks for a British commissioner to be proposed, accepted and vetted by the European parliament.

Heightening the pressure on Johnson’s government, von der Leyen has called on the UK to propose female candidates, as part of her bid to have a gender-balanced commission. If she insists on that point, the UK’s current commissioner, Sir Julian King, would be ruled out. But any UK commissioner would have to stand down on the next scheduled Brexit day, 31 January, unless another delay is agreed.

King, a career diplomat, was drafted in by David Cameron when the UK’s previous EU commissioner, Lord Jonathan Hill, quit in the turmoil following the 2016 result.

The commission is a civil service with political leadership: at the top are 27 commissioners and one president – one per member state. Von der Leyen has four vacancies, including that of the UK, but hopes to fill the gaps rapidly, allowing her to take office on 1 December, one month later than planned.

A UK government spokesperson said: “We have received and are considering the president-elect’s letter. We will abide by our legal commitments.”

The demand on London emerged as the outgoing head of the commission said Brexit had worsened Britain’s problems, in another parting shot at the government.

Jean-Claude Juncker delivered a typically blunt assessment in an interview with ARD, the German public broadcaster. “Britain also has problems other than Brexit and these problems have got bigger with Brexit. They are trying to cover that up, but they just got bigger,” he said.

Juncker also described the UK’s departure from the EU as a perpetual cycle of “promises, promises not kept, and lies repeated over and over again”, two days after he accused Boris Johnson of telling “so many lies” during the 2016 referendum campaign.

In contrast, Juncker struck a more diplomatic note in an interview with the BBC, saying he liked the prime minister.

The veteran EU leader did not elaborate on what he perceived as lies. He may have been referring to the NHS spending pledge on the Vote Leave campaign bus that was described by the UK Statistics Authority at the time as “misleading” and undermining trust in official statistics.

He may also have had in mind another central claim of the leave campaign that Turkey and four other countries could join the EU by 2020, a statement not supported by anyone who follows EU enlargement. Juncker announced a five-year freeze on EU enlargement in 2014, while Turkey’s accession hopes were on the rocks long before the 2016 referendum.

In a blow to Labour, Juncker described Jeremy Corbyn’s party’s plan to renegotiate the Brexit deal as unrealistic, although he also noted it would be up to his successor to decide if there were “room for manoeuvre for a new deal or a new treaty”.

Privately, sources appear more receptive to a request from Labour to renegotiate the political declaration on the future EU-UK relationship. The non-binding declaration, which is Labour’s biggest problem with Johnson’s deal, points to a free-trade agreement that it claims would reduce GDP per capita by up to 7% within a decade. One diplomat recently said it could be rewritten overnight.

The commission president also expressed scepticism about Conservative hopes of negotiating a free-trade deal in 11 months, telling the BBC it would “take time”.

The EU’s chief Brexit negotiator, Michel Barnier, has said a basic free-trade agreement could be reached within 11 months, but he also warned the negotiation would be “difficult and demanding”.

Brussels is additionally unconvinced by Johnson’s pledges to avoid extending the transition period beyond 2020. Barnier said the summer of 2020 would be “a moment of truth” to see whether an extension of the transition will be needed.

Juncker, a veteran political fixer with four decades’ experience in Luxembourg and Brussels, has made several key interventions during Britain’s departure, most recently sealing the deal with Johnson over the course of two phone calls on 17 October.

He will be succeeded by Von der Leyen, whose start date has been delayed due to problems having her team approved by the European parliament. She is meeting Tony Blair in Brussels on Wednesday at his request. The commission and the former Labour prime minister’s Institute for Global Change declined to comment on details of the meeting.