Have you cleaned the heatsink and fan?

@Blackwater_Park I looked inside the case when I changed the CMOS battery, and it all looked pretty clean. As I said, there is no problem once it’s running, only on boot-up and not every time, so it isn’t an overheating problem.Thanks for your answer.

If your PC is from 2018 Im guessing the CPU (Intel 7800x?) is probably a bit too weak to fully take advantage of an RTX 4070. Also, depending on your monitor and refresh rate, you may be able to spend less and get the same real world performance. You may be better saving the £500 and putting it towards a new PC. Also Nvidia is expected to announce RTX 5000 soon and when they do you may see a price drop on older models as retailers try to move last gen stock so may be worth waiting?
Finally the fact that it works without issue on the second attempt is a bit weird.

The CPU bottleneck thing is not really that important.
I would be more concerned about the fact that Dell uses weird proprietary power supplies.
You should check if it has enough PCIe power connectors for the replacement card, that the power consumption for the new card is similar to the old one, and that the card actually fits inside the case.

For example, the 4070 that I use should be small enough to fit inside the case, but the 4000 range has a new proprietary power connector. It comes with an adapter, which requires 2 full PCIe connectors from the PSU.

@NovDel Before you buy a new card, power down your PC, pull the card and clean the heatsink, fan, and socket. Remove any internal unused peripherals from the power supply. You may find that the boot-up beeps go away.