What you have completed so far is an "Agreement in Principle" (AIP).
The AIP is an initial credit and affordability check on you based on the basic information that you would have provided the broker - name, Address, DoB, past addresses for last 3 years, Income, financial commitments, dependants, knowledge of adverse credit.
With this info the lenders can trace you through the credit reference agencies and see what you have been like with your credit. They can view all of your recent payments, to who, how much, if you have missed any payments or even just been late in the last 12 months, how much credit you have available, if you have ahd any defaults or ccj's or anything more serious in the last 6 years.
Based on the income declared and your current credit commitments you have in place the lenders will apply their affordability calculations to your application and make a decision on whether they are happy to accept you as an applicant or not. If they are happy with your credit score they will either diclose an amount that you can borrow (could be above or below what you want to borrow) or agree to the amount you want to borrow (depends on the lender).
Sounds complicated but 99% of the work is done at the lenders end, all you have to do is give the broker the information and then a result will come back which is generally pass or decline. A decline means you cannot submit an application, a pass means you have the green light to submit an application.
Once you have a pass from the AIP you have done the hard work, all that remains really is for you to submit the application which will give the lender all the details about the house you wasnt to buy, who from, how much for, what deposit you have, solicitors etc. They will also produce a list of documentation that they want to support the application. The documentation is normally requested to evidence the information you disclosed in the original Agreement in Principle (AIP) and also the application (which is based on info from the AIP. Assuming all your payslips agree with the previously disclosed information and your credit is good and you do not have too much in the way of financial commitments like credit cards or loans etc then the lenders underwriters will most likely pass you as acceptable borrowers. After this then you have the property to be surveyed. Assuming that a surveyor find no problems with the property then he will send a report back to the lender with a valuation based on the condition of the property, saleability and any potential work that may need to be carried out. If the lender is happy with this report and if the property is valued at the asking price or above then the lender will then issue a formal mortgage offer and from here you then have the legal work to be completed.
Cut a long story short.
From AIP to decision is about 30 seconds. From Application submission to acceptance is normally 5-7 days. From acceptance to valuation of property is around 3-4 days. From valuation to mortgage offer is 5-7 days. From mortgage offer to completion of legal work and money drawn down is about 3 weeks. This would make the total time for the application to moving into the house about 6 weeks if things go relatively quickly....it can be significantly longer and it can be shorter but even at best i would expect 4 weeks.
Good luck and give the fingernails an chance!