Although the bank may allow you to increase the mortgage on the first property so that you get cash out immediately, they may ask what it's for and I thought there was a general reluctance for banks to give someone a mortgage if they have to "borrow" for the deposit. However, if you have equity, there's no time limit/minimum period before you can release it as far as I know. The bank may be sceptical about the 250k valuation and will have that independently verified.
In a best case scenario, let's say you can liberate 47k from the existing house (leaving you a 90% mortgage based on the 250k valuation), they will need to see that you can pay back the mortgage on the original house with the rent - they assume (i) 10 months' occupancy in every 12 (ii) 10% maintenance fees per annum and (iii) a "stress test" to ensure that you can make repayments out of that money if the interest rate goes to 6.75% (AIB picks this rate anyway - don't know what other banks use) - at a borrowing level of 225k, your repayments will be in the 1200 per annum region (you can check exactly using any of the mortgage calculators) so you'd want to be renting the place out for prob at least 1400 per month.
You gotta talk to the bank - they will tell you soon enough if you can do what you propose. I'd say there will still be houses out there after the week it takes you to arrange to meet them! I found AIB very good for getting me a good idea about these things (rental returns, quick overview of how much you can borrow); although I probably won't go with them because of higher interest rates.
Sprite
