Brexit passport renewal - Will I need to pay again?

The answer is almost certainly #1: it's extraordinarily unlikely that existing British passports would be completely invalidated by Brexit. A British passport states that you're a British citizen, which will continue to be the case even if Britain leaves the EU.

While I can't point to any official sources, I can note that a) passports of European countries issued before they joined the EU remained valid even without the EU designation, and b) passports of countries that ceased to exist, such as the Soviet Union, still remained valid afterwards. (It took Russia six years after the dissolution of the USSR to even start issuing Russian passports!)

In your comment, you linked to an official source that (rather understatedly) answers your question:

To save the taxpayer money, the newly designed passports will be introduced in a phased approach.

After the UK leaves the EU, burgundy passports will continue to be issued but with no reference to the European Union.

New blue and gold passports will be issued from October 2019, when the new passport contract begins, to those renewing or applying for a new passport.

There is no need for British passport holders to do anything ahead of their current passport renewal date.

(Emphasis added.)

Of course, the first paragraph is somewhat disingenuous:

After Brexit, the UK travel document will no longer be required to conform to EU standards. So in a move to symbolise our national identity, the cover will be changing from the standard EU burgundy colour to a blue and gold design.

This implies falsely that the burgundy cover was required by EU standards; in fact, it is just a suggestion, and the UK could have gone back to a previous passport color, or indeed any passport color, without leaving the EU, just as Croatia has retained blue passports after joining it.

The implications for your question:

I've read/heard that when we leave the EU, also destined for March 2019, this will result in passports needing to be replaced with non-EU ones.

That is incorrect. Existing passports will remain valid until they expire.

Has any advice been published on when/at what cost this replacement will happen?

No, because there will be no such required replacement. People who want to pay the fee to renew their passports early will presumably be able to do that.

Some scenarios I've thought about; 1. If I renew now, the new one will stay valid until its expiration; new "British" passports would be issued only as existing "EU" passports expire.

This is what will happen.

2. I renew now, and in 5 months get issued a new "British" passport, without extra charge.

That is wishful thinking, but fortunately it won't be necessary to get a new passport.

3. I renew now, and in 5 months my "EU" passport dies, and I have to pay for the new "UK" passport.

No. The EU doesn't issue passports. An "EU passport" is a national passport issued to an EU citizen by an EU member state. When the UK ceases to be an EU member state, UK passports issued to British citizens will cease to be EU passports, notwithstanding any references they may contain to the European Union. But they will continue to be UK passports, and valid.

4. I don't renew now, and wait for the new "UK" passports to be announced and get one then.

They've already been announced, as noted above, but you can certainly wait until October 2019 before getting your new passport, if your travel plans allow. In determining whether your travel plans allow, don't forget that some countries require extra validity beyond the end of your visit.

If you don't need it soon, leave it until it's about to expire. This is because the government have recently changed the rules and no longer add the additional time to the expiry - so if you applied now (October 2018), your new passport will expire in October 2028, not March 2029 as it would have under the old rules.

They've done this because the Schengen rules require passports from third countries to have been issued within the last 10 years, and if you've got a passport valid for 10 years and 9 months (as was possible under the old system), you'd not be able to use it to visit Schengen for the last 9 months.

See this gov.uk page for more details:

Since 2001, some adult British passports were issued with a validity longer than 10 years. This is because if you renewed your passport before it expired you were allowed to have the time left on your old passport added to your new passport. The maximum validity period possible was 10 years and 9 months. This means you can’t use the expiry date to check if your adult passport will be valid under the new rules.

From the beginning of September 2018 extra validity is no longer added to passports and the maximum validity for a new adult UK passport will be 10 years, and for a child passport will be 5 years. We have made this change to follow recommendations set out by the International Civil Aviation Organisation and to help provide clarity about passport validity in the Schengen area in the future.

What's more of an issue is the possibilty of needing some number of months' validity for visits to Europe in the future (unlike now). Many countries require 6 months from date of entry, or something similar. This applies in particular to the EU (gov.uk). This doesn't affect you directly as your passport runs out almost exactly on brexit day. the rule does mean that a surge in passport applications can be expected, starting imminently. This is likely to mean increased lead times around the time you need to renew yours.

I renewed my British passport just a few weeks ago using the online service. A few days later, my new burgundy passport arrived with an expiry date in Sept 2028. I have no doubt that it will remain valid for all travel even after 29 March 2019.

The only change that I am aware of, is sometime after Brexit the UK passport office intends to start issuing passports with a blue cover instead of the current burgundy colour.

Mine was due to run out on March 2020, I went to the post office a week ago and paid £90.40 for the renewal, it arrived yesterday 30th October 2019, the words European Union had gone but it was still in Burgandy not Blue, they’re using old stock until the beginning of the year.

Most of the answers here consider this topic from an entirely UK-centric view. For its main use, it is relatively irrelevant whether British authorities consider a UK passport with "European Union" on the cover valid or not.

Where you need a passport is entry and exit to another country. It is very unlikely, but you could easily argue EU countries would consider the "European Union" cover as untrue and thus deny passport validity (There have been lots of similar examples in the past). How this will be handled in future is not a simple UK decision, but rather would be one small part of the EU/UK series of treaties that need to be sorted out between the UK and the EU.

Current passports will stay valid for ten years after they are issued. The only difference: With current rules, if your passport expired March 2019 and you renewed in January 2019, you received a passport valid to March 2029. Now it will only be valid for ten years, that is January 2029.

What you need to decide is whether you want an old passport for the next ten years, or if you want to stand out in every passport queue as a Little Englander with then "new" (1970's style) blue passport. If you want to get the old passport, renew before March 2019. If you want the new passport, renew after March 2019. This may mean that you are without a passport for a short time, or if you have a brand-new old style passport and want a new-style one then you'd have to pay.