Should I take a gap year before an engineering degree?

I don't know how rigourous the courses you are considering are but it depends a lot on this. If you have already applied, it may be worth contacting placed you get offers from on whether you can defer. For really rigorous courses, I would probably mention how you will keep engaging with your subject e.g any maths/engineering books you will read. Any podcasts you'll continue listening to etc. if you are yet to apply you have two choices: apply, get your offers then do what I've said above OR mention you are taking a gap year, apply for deferred entry and explain how you intend on engaging with engineering in your gap year.
I think places are only really hesitant for a pure maths course and even still, they often allow gap years if you say you will continue engaging with the subject. I advise you actually do, just so you can keep up with the course otherwise it will be that little bit harder when you start. I've heard some places may ask you to evidence your engagement.
If you want to take a gap year, do it!!!! I doubt it will affect your offers if you do the above. I don't know how imperial would respond so email them to check this is okay, but for Oxbridge I think this would be sufficient and remember gap years are common for many courses even at competitive unis, especially as some people reapply, so don't worry too much

My sister lives in Hong Kong and she has given me the opportunity of a lifetime by offering me to stay in her spare room whilst I take a gap year before uni. Normally I would jump at this opportunity but I know that uni’s tend to dislike it for maths
based subjects because they believe students get rusty and I’m wondering whether to throw away this opportunity and potentially regret it forever or to go straight into university with a higher chance of being accepted. PS I am totally open to planning a gap entered around engineering although I don’t want to do a 12 month long work placement like lots of uni expect from you. Any advice is massively appreciated!

Have you already applied? If so you'll need to check if the unis you've applied for 2026 entry to are happy for you to defer entry for a year. Many likely would but you'll need to check in any case. If you've not applied yet then it's totally up to you.
The gap year maths admissions thing is really specific to a) mathematics degrees and b) Cambridge (and not even all colleges at Cambridge).
There is a practical element to consider which is if you've not solved an integral for ~12 months you might find it a bit of an adjustment getting back into it and may have forgotten some useful skills, but that's more in terms of actually succeeding on the course, not in terms of admissions. Also there's no reason you can't just do a couple practice problems every day during your gap year to keep on top of things (you don't even need to do intensive revision or anything, just a little bit of something regularly to keep yourself sharp).