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Flow of money on refinancing?


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Hiya!

I'm struggling to understand the flow of money when refinancing a property, which I'm sure is simple to most of you ūüėõ¬†!

Can anyone help explain if this example is correct?

Stage 1 - Buy property
PP: £100K
Deposit: £25K (your money transfers to lender)
Loan: £75K interest only mortgage (75% LTV)

Stage 2 - Renovate
Say you spend £10K of your own cash renovating

Stage 3 - Refinance (say at end of 2 yr mortgage fix)
Reval: £120K (£20K value uplift)
Outstanding loan: £75K
Current equity: £45K (Reval - Outstanding loan)
Remortgage (at 75% LTV) loan: £90K
Cash out: £15K (The incremental difference between current loan and new loan e.g. £75K existing, £90K new value, means £15K additional loan payable in cash)
New equity: £30K

Two questions;
1) Does this mean the lender essentially pays you £15K in cash upon remortgage? 
2) How does the paying you cash scenario work if you change lender? Flow of money etc...

Appreciate help from anyone explaining this for me guys!!
thanks 

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