I've been in similar situations, but there is no ideal situation. Suggestions are -
1. Zero chance of getting results early - there's a quasi legal process and the Uni can't and won't budge on that.
2. Have a discussion with your current landlord about handing in your notice - while they will obviously not want a gap in tenancy, demand is high in Bristol and they might be prepared to accept your notice, go slow on refilling, and re-accept you if the move doesn't come off - if you've been good tenants, that is a better option for them.
3. Build relationships with Letting Agents in Oxford, stressing your 'maturity' not your student status. Be clear about your need for an established, 'family' type accommodation suitable for a cat. They will have a much more nuanced knowledge of landlords and who will flex the 'no students' and 'no pets' rules for the right sort of tenant. I think you will be surprised at how many will.
4. Storage companies like Yellow Box usually have quite significant discounts for the first 6-8 weeks, and you can negotiate a bit, so storage, while not good money to be spending, might be less that you think.
5. Don't mention that cat, and if it gets found during an inspection, pretend it's an emergency rescue from a family member in hospital - but I didn't say that!
6. You could look and see if there are any affordable monday to Friday rooms available in Oxford, but :/