The tax doesn't matter so much with the lower CO2 band vehicles as it's only £0-£30, but for yours it's probably something like £145 a year. When doing a first registration & taxing on AFRL (Automated First Registration & Licensing), for tax bands over £30 the dealer can add 'extra weeks' so that if they're registering part way through a month, the expiry date of the tax will still be the same as if they taxed it from the start of the next month (if that makes sense), so in theory your tax should still expire at the end of March 2017, like it would do if they'd taxed it during April, so hopefully you won't need to tax it a month 'early' next year.
The finance in your case also shouldn't matter but I was concerned that cctesta had already made the first finance payment. It sounds like they've held-off collection due to holidays, which is fair enough, but I wouldn't usually expect the first finance payment to take place till about 2-3 weeks after collection of the vehicle. If any concerns regarding this I'd contact the finance company who are more than likely FCA Automotive Services (formerly FGA Capital).
It's the warranty that bugs me most as, like you say, it'd be typical that you need to submit a claim towards the end of the warranty (after taking the car for a first MOT for example) and whilst Fiat can show some flexibility, they're not obliged to help once the warranty has ended. Obviously if it started before you were using the car, this isn't really fair unless in the case of things like pre-reg or other stonking deals where the price paid makes up for the lesser level of cover.
Anything up to a week or two I'd just about put up with, but anything longer than two weeks I'd be onto Fiat UK & say you want the warranty extending or another form of compensation (ie first service free) to make up for it. It was a factory order, and they did you no favours on the trade-in value for the Panda, after all (even if you decided against trading it in).
Best of luck!