The main focus of the delays centre around AstraZeneca. Pfizer has also had delays, but they are less relevant.
The claim by the EU is that the contract they have with AstraZeneca is that their manufacturing facilities in the EU and the U.K. should all be used to ensure delivery of the initial doses (300 million + 100 million) that the EU ordered.
The claim by AstraZeneca is that the contract is a best effort, not a guarantee of delivery. They have argued that the reduced yields seen in the EU are because the EU signed the contract 3 months later than the U.K.. Therefore, similar teething problems were ironed out much earlier - before the vaccine was due delivery (even before it has passed the clinical trials).
The full contract is available here, but most reports (1, 2, 3) I’ve seen are that the contract appears to support AstraZeneca‘a stance:

AstraZeneca says that the U.K. has priority on the vaccines manufactured within the U.K., and that the U.K. site is not included in the primary manufacturing centres for the EU - and so cannot be expected to be used to make up the shortfall.
However, the U.K. also relies on the Pfizer vaccine, which is manufactured in Belgium. The EU, by bringing in the export controls, can potentially limit the exports of the Pfizer vaccine, to make up for the shortfall in the AstraZeneca one. Whether they will do this or not, remains to be seen.
The U.K. government has not commented on this specific matter, other than to restate that supplies of vaccines to the U.K. are secure. Again, whether this is the case remains to be seen, but given the AstraZeneca vaccine (and the yet to be approved Novavax vaccine) is manufactured in the U.K., it seems plausible.
The EU has been reinforcing the fact that it invested billions in vaccine development, and that they now expect the firms to deliver. While I obviously don’t think spending more money should be the only criteria for receiving vaccines, the FT reports that the EU has spent about 1/6th of the amount per capita than the US or the U.K. has - despite ordering slightly more doses per capita. This, combined with the late order date, likely isn’t helping matters.
In another development, the EU invoked - and then backtracked on invoking - Article 16 of the EU-U.K. “Brexit deal”, which imposed a hard border on the island of Ireland, the land border between the U.K. and the EU. This received widespread condemnation within and outside the EU, in a rare case uniting people who were both for and against Brexit! This was, the EU claimed, to prevent vaccines leaving the EU via the U.K., in avoidance of their (potential) export controls.