They are not a minor inconvenience, but neither are they an insurmountable obstacle. The question is what price are you willing to pay?
This is expensive, and has a high maintenance cost.
Regauging would be difficult due to the need to maintain the other Irish gauge lines. Dual gauge is possible but carries a maintenance penalty.
A new HS line is probably the optimal solution, but it would be difficult to see a positive business case.
'direct' is a generous description of London to Belfast/Dublin via Stranraer. Edinburgh-Dublin is a tiny market: 650k annual air passengers Edinburgh-Dublin against almost 4 million London-Dublin. And don't expect all of those to transfer to rail.
Before the single market, this was common all over Europe. It could be as easy as the Irish customs sealing the containers before the train departs, and then the French customs checks the seals when the train arrives.
Direct as in "no need to change train".
If you add Dublin-Glasgow, Belfast-Glasgow and Belfast-Edinburgh you get close to 2 million pre-Covid passengers. That is on average 5250 daily passengers, or enough to fill almost 12 daily ICE3s.
Spain has dual gauge high speed trains which would resolve the gauge problem.
The services would be something like:
Edinburgh
Glasgow
Liverpool
Manchester
Birmingham
London
Belfast to Dublin would need to be rebuilt to provide viable journey times. Ironically for a unionist project, it would rely on most services continuing to Dublin to have a half viable business case.
With some upgrades in Cumbria London to Belfast in 3.5 hours and Dublin in under 5 would be viable through a high speed line between Belfast and the WCML. The former would take most of the market but the latter would be niche, maybe a one morning and one evening service per day. Birmingham would have more competitive journey times but is a much smaller market.
Manchester and Liverpool would be more competitive and pre pandemic had multiple flights to both. Glasgow and Edinburgh to Beflast and Dublin would have very competitive journey times.
I reckon the passenger service could equal the Channel Tunnel:
1tph from London to Belfast by extending HS2 service that is due to terminate in Lancashire. Would also serve significant parts of the North West. 2tpd extended to Dublin.
Shorter HS sets:
1tpd from Birmingham Curzon Street to Dublin via Belfast.
1tph Scotland to Dublin service, alternating between Edinburgh and Glasgow and becoming the main Belfast - Dublin service.
2tpd from Liverpool to Dublin via Belfast
2tpd from Manchester to Dublin via Belfast
That would be a total of about 37 services per day each way through the tunnel. It would need solid freight traffic to be half viable.
Its not as mad as its being made out to be but it would be a political not economic project. Even if the UK government decided to fund it, I doubt it would get the legal guarantees from Dublin and Edinburgh that it would need to avoid picking up the bill in event of Irish unity and / or Scottish independence.
That is probably a better idea.
But then there is the big question of how much this will cost, it doesn't sound like a cheap proposal.
6 trains each way doesn’t seem like a market to build a new railway for.
It would clearly be a political project. The obvious (political) solution would be to extend HS2 services (e.g. Preston-Lancaster terminator) and run services from smaller cities with shorter high speed trains comparable in length to 5 car 800s. It would need to part of a HS cross country network to make any sense.
As the earlier poster alluded to, surely the true market is/was Dublin - London, which, along with Amsterdam - London, is/was one of the busiest air travel markets in Europe.
Given that HS2 was being built as far as Crewe, an upgrade of the North Wales Coast Line, and a tunnel beyond Holyhead, combined with a HS2 - HS1 link, could have given us a Dublin - London - Amsterdam (- Berlin).
If only we hadn't have had the Tories in 1992, imagine how much further we'd be now, instead of going further backwards!
Dublin to Crewe would require a tunnel about twice as long and a rebuild of the North Wales main line. Possibly cheaper but an enormous project.