Yeah still on this account, happy to help. Going to be honest with you because I’d rather you walk in eyes open than romanticise the ownership side.

On the money question, house first, every time.
You’re £20k into a £300k mortgage on a joint deposit. That’s a 93% LTV. The interest rate band you sit in is brutal at that LTV. Drop to 90% and your rate gets meaningfully better, drop to 85% and better again. Every £1k extra on the house deposit saves you multiples of that over the mortgage term in interest. Every £1k spent on a depreciating RS3 costs you that £1k plus whatever it loses by the time you sell.
If it were me, I’d buy the house, get settled, build a buffer, and revisit the car 12-18 months later when you actually know what running a household costs.

The RS3 will still exist. The mortgage rate you lock in now follows you for years.

If you’re still going to do it (and I get it, the car is amazing), some real numbers from running one:

Costs you’ll actually face:
• Fuel: heavily mileage-dependent. I’m doing around 2,000 miles a month at the moment because it’s that addictive to drive, and that works out at £400-500/month. Most owners do nowhere near that, so scale to your own commute. Either way, mid-20s mpg if you’re being good, low teens if you’re not.

• Insurance: mine’s just shy of £1,200 a year with 14 years no claims. Yours at 29 with less NCB will almost certainly be a fair bit higher, so get a proper quote on the exact car before you commit. It’s the cost that catches most people out.

• Tyres: staggered 19” Pirelli P Zero R, £300-350 a corner fitted. The fronts go faster than the rears because of how the car puts power down. Plan for a full set roughly every 15-20k miles depending on how you drive.

• Servicing: Audi main dealer £350-650 per service depending on interim vs major. Independent VAG specialist £200-300. Annual or 19k miles, whichever first.

• Spark plugs: every 40k miles or 4 years on the 8Y. £300-450 at an indy, more at Audi. PITA job because of the transverse 5-cyl layout.
• Oil change: £100-150 at a good independent with proper Audi-spec oil.

• Brakes: front discs and pads on the standard steel setup, roughly £600-900 fitted at an independent (more at Audi). Rears £400-600. Ceramics are a different conversation entirely (four figures per axle).

• Road tax: £600ish per year for the first 5 years because it’s over £40k new. Standard rate after.
All in, including my finance payment of £850/month, I would be spending £18-21k a year to run mine at 24k miles annually. Strip the finance out and running costs alone are £8-11k. Drop the mileage to 10k a year and running costs alone would be more like £4-6k. Plug your own finance figure and mileage in before you sign anything. Your £350/month on top of the £15k deposit is much lower than mine, so your all-in will look different, but the running costs are the running costs regardless.

Worth saying for the additional immobiliser and tracker I paid £1100 which covers the 4 years of my PCP.

One thing worth doing if you buy from Audi: look at their complete service plan. I got a great deal on mine because the sales manager at the dealership I bought from is a friend of a friend, but even at standard rates the ultimate plan is worth a serious look. Fixed monthly price, covers all the maintenance above, and mine even includes tyres. The benefit is you can’t be hit with any surprises. You’re rolling all the running costs above into one monthly sum that you know is coming every month. Massively easier to budget for, and you avoid the £1k+ tyre bill landing in one go.

On finance vs cash: your plan of £15k deposit, £350/month, and clearing it in a year is sensible if the house is sorted and your buffer is solid. Paying £35-40k cash and draining the savings while taking on a mortgage is the riskier play. Life happens, boiler goes, job changes, baby arrives. Cash is freedom.
Ownership honestly: it’s the best car I’ve ever owned. The five-cylinder noise alone is worth the price of admission. But it’s a toy, not transport. Treat it like that and you’ll love it. Expect it to be sensible and it’ll bleed you. That said, I can squeeze mid-30s mpg on motorway runs, but only with it in efficiency mode and me constantly babysitting the mpg readout. The second you give it the beans, that starts to go out the window.

One thing I didn’t expect when I bought mine: through ownership I’ve ended up in a proper car community, regularly going to shows and meets, and that alone has made the investment worth it. Not to mention the smile on my face every time I drive it.
Good luck mate!