Going into a data science career after a BSc degree in Physics and Maths.

Keep doing what you're doing. You need to be fluent in python and it would also be beneficial to learn SQL and R. It's important to be familiar with as many data science libraries as possible. This includes numpy, pandas, matplotlib and scikit-learn. It would also be good to familiarise yourself with some of the more popular ML techniques. Some companies are looking for candidates that have experience with NLP for example and in particular they're looking for people who have some familiarity with PyTorch and Tensorflow. Other than that, you just need to be able to work part of a team and also be able to work independently. It's not that hard. This just means that you need to be able to listen and contribute whenever you can and try to learn from your mistakes.

I'm 27, and I currently work in ML engineering and consulting for household-name companies. I was in a similar position to you when I was 19, and so I think I am qualified to give some advice.

In terms of academics, you're doing just fine. Ensure you get a First, learn Python and SQL, and keep plugging away on Kaggle. However, the main thing you're going to need in order to be successful is data analytics skills.

Data science is a fantastic field and is truly fuelling the future; it is your job as a data scientist to connect your knowledge to actual problems that people want to solve. This requires that you have a keen eye for data storytelling: you know how to turn insights from data into a narrative that will convince stakeholders to take action (usually that action is to invest time and money to build an ML pipeline and spend money training models ) You won't be expected to know everything about the latest transformer architectures as an intern - they will be looking for someone who is keen to engage with the actual problem of solving real-world problems using data.

tl;dr build some data dashboards on Tableau Online or Power BI which use the outputs of some basic ML (logistic regression, L1/L2 regression, k-NN etc) to tell a data-driven story. Use this in your interviews.