I would do some research into GCSE requirements for vet schools if you haven't already. Some GCSE requirements are low (e.g. Bristol at least a 4 in English and maths) or aren't even mentioned - you could contact vet school admissions too, to ask about any specific GCSE requirements. There may also be some leniency when you apply due to the events of this year (but again, admissions teams will be better to answer this).
Getting good A level grades and your work experience is definitely a good aim!
Yes you can apply for zoology and veterinary medicine at the same time. When you apply to uni you have 5 choices for universities/courses, up to 4 of which can be for veterinary medicine. Vet applicants often apply for a 5th choice in a similar field (e.g. zoology, bioveterinary, gateway courses if you're eligible, etc etc) to use as a backup in case they have no vet offers or miss their grades. However, many ignore this 5th choice with the vision that they would take a gap year and reapply if they got no offers. There is also the option to resist A levels and apply with better grades, although you would need to pay attention to which universities accept resits. Veterinary medicine is competitive and even good applicants may not get any offers the first time around. It is for this reason that even though you could, in principle, apply for 3 vet courses and 2 zoology courses, I would encourage you to apply for 4 vet courses.
You understand best your abilities in getting your grades, so it's up to you how you go about it. Carefully check the GCSE requirements and think if you will get the required grades at A level. If you meet the requirements then I would advise that it's best to do veterinary medicine as a first degree. There are also gateway courses that you may be eligible for. Then go for graduate entry when you're certain that you're not going to meet the grades.