I can answer a few of those questions.1. Is it best to study a degree with 1 job sector or career in mind or is me trying to keep my options open a good idea?
Probably to keep your options open. It's my motto, but at the same time it depends on what the entry requirements are for those specific industries/careers.
As far as I know, you can go into quants with an engineering degree, but you will struggle to get into an engineering grad scheme with a degree in finance (or financial mathematics). So I would play it safe and go with a degree in electronic engineering.
A degree in robotics isn't necessary going to limit your options that much, but it seems like a very niched field i.e. you don't normally get a lot of employers or professional qualifications asking for degrees in robotics. It's more likely you will get people asking for degrees in electronic engineering.2. How can I find a course that applies to all my interests?
You can't as far as I know. If you're looking for something that involves robotics, electronics, and mechanical engineering, you might be able to find something in the engineering field. On the other hand, it's incredibly unlikely that you will find something that would marry both engineering and finance in one degree. It's likely that you will need to do at least 2 degrees/qualifications in order to meet this requirement.
I don't know how you intend to become competent in quants, but you can consider the following:
A degree in financial mathematics with a lot of programming
A professional qualification recognised by the iindustry such as the CQF (highly recommended)
Get training on the job (if you can secure the job)
The routes into robotics are generally:
Research in academia (if you are going for a research role)
My plan of action would be to do an undergrad in electronic engineering (or something along those lines) and then do a CQF. If you prefer getting a master's degree (I can't quite understand why, but whatever floats your board), they would normally ask for people with a quantiative background e.g. an engineering degree3. Can I get into the quant sector by doing engineering degrees like I have linked above and then doing a financial engineering masters?
Yes. I think most of the employers care about whether you have the numeracy and programming skills. These imply that you have done a degree in some mathematical discipline e.g. engineering, physics, maths, financial mathematics (maybe).
In terms of doing a financial engineering degree, you will need to check the entry requirements of the specific degree. For example you should be fine with the following:
You are unlikely to be OK with the following:4. What are the best courses I can study to get into the quant sector?
Hands down, CQF. There are master's degrees available, but I consider the material a bit too theoretical. CQF on the other hand uses more practical application.
Possible key words that you might want to look for can include:
A lot of the degrees for the above would be at master's level. Financial mathematics do have some degrees at bachelor's level though.
Do note, you can go and do a postgrad degree in something related to finance with an undergrad in engineering, but you can't go an do a postgrad in engineering with an undergrad in finance.
See the following for how to become a quant:https://uk.indeed.com/career-advice/finding-a-job/how-to-become-quantitative-analyst
You would probably want a second opinion from someone who works as a quant regarding the above.