I don't think there is a best time to start, do it when you are ready.
I'm also in the UK and the process via the NHS is very slow and it's expensive if you go private. In addition to that there are changes from HRT that become less likely with age. I would advise you to contact your GP and ask for a referral to a GIC, ideally one you will easily be able to access after Uni. There will be a waiting list and it's likely to be a year or more. There's no risk to this, if you change your mind you can cancel the appointment and you don't need to tell anyone other than your GP.
If you can afford it I would have a chat with a gender therapist.
Employment is a bit tricky as you didn't say which sector you are interested in. I started to transition at 41, with a well-established work history and qualifications and providing references and certificates is going to be a challenge. Sectors such as IT and Finance are considered to be some of the most accepting, try to find work in a city if possible. I would guess acting and the arts is good too. I've found that with a positive attitude and some resilience finding work is not a problem so I tend not to worry about work and money to much.
BTW, you don't have to present as male or female regardless of what the NHS push, you can experiment and make changes as you will. Perhaps try small changes and see how people react. I started carrying a gender appropriate bag and other small changes long before I was out at work.