Can I apply for UK Visa two weeks before travel date

Can I renew my uk visa for 2 years?

Starting with an important technicality, despite the common vernacular of "renewing", there is no such process. Every application starts from square 1 and is examined on first principles. Somebody can get 3 or 4 visas back-to-back and start to think the process is a formality. This can lead to disappointment, so don't think of it in terms of "renewing", it's a "fresh application" and everything needs to be spit-spot. Rule of thumb:

Where UKVI is involved, nothing, *ever* is a formality.

You want to know if you can apply for a two year Standard Visitor Visa. This is always a possibility (the rules permit a person to make an application for anything they want), it's simply a matter of checking the two year box and paying the corresponding fee. The real question is how it will be treated; they look for three things...

1. A lengthy history of performance; it means all of the explicit terms and conditions have been observed. They will also look for any dodgy history like where you told them you were going to stay for two weeks and ended up staying for six weeks (even though you didn't overstay). Obviously, first-time applicants second-time, third-time, etc) will fail on this point because there's no history of performance.
2. Activities while visiting; it means your visits do not indicate that you are building up a private life in the UK. If they get the idea that you contemplate making the UK your home, there will be problems.
3. A viable premise (this is the most important one): it means they will not issue a longer term visa when it's a matter of convenience for you or just because you want one. You need to show that your lifestyle is stable enough to predict the need for a series of visits looking forward. A good premise, for example, is a grandmother with grandchildren in the UK who has a lengthy history of family visits.

All this, plus the application is diligently prepared with respect to all the suggested evidence (and that means hard evidence).

When you submit the application, several things can happen...

• It can be refused on one of the points above; or
• It can be approved, but for the standard term of six months (and they keep the extra fee you paid); or
• It can be issued for two years like you asked for.

I have also seen where the ECO 'automatically' upgraded an application and issued a longer term. In that case they will get in touch with you and say something like, "...although you applied for a standard six month visa, the ECO has determined that you qualify for a two year visa, would that be acceptable?...".

Even though I plan to travel in 17 days time? Should I state reason for Late Application?

The link provided by 'pnuts' (to whom thanks) will level your expectations about how long it will take to process your application.

I also plan to use Priority Service!

The priority services are great when you qualify to use them, but they will not help make the case for a longer term visa, it just moves you to the top of the queue. As I wrote above, they will start from square 1 each and every time.

Should I include in a cover letter, reason for late application?

Is it important to make that kind of explanation? Or will your letter end up on the mail room floor after it's scanned by an assistant? You can also use the free text 'Remarks' section at the end of the application if you think an explanation is necessary (personally I do not). You need to spend more time polishing your premise. Having said that, a high quality cover letter will be passed to the ECO and may carry the application. Professionals work hard on cover letters and ECO's appreciate them.

Finally, nobody on the net can give you an accurate prediction of what will happen. In order to do that you need to get a qualified person to examine all your evidence and understand your personal circumstances.