How important is credit history when applying for a mortgage?

I have £20,000 to use for a deposit towards a mortgage on a property, I am looking at properties in the £200,000 range with a yearly salary of £60,000.

I have minimal credit (a credit card that I have no late payments on in over 3 years, bank account with no overdraft) and all my other accounts paid in full over the last 5 years (mobile phone, utilities) and I have been on the electoral register for the last few years.

Around 2 and a half years ago I took out a few payday loans (the most recent is 30 months ago) and paid them all back on time, this was due to being temporarily unemployed and experiencing a family emergency although I don't think the reason matters. My finances are now stable and have been since I got my current job (salary started at £50,000 per year 2 and a half years ago, moved to £60,000 about 10 months ago).

Will having these payday loans in my credit history make me ineligible for a mortgage? Should I wait another 3 years for them to disappear from my history (because of the 6 year expiry)?

Another question is I was reading about the "Help To Buy" scheme, would my financial history make me ineligible for that scheme?

Thank you in advance.