Hi, and welcome to the forum.

Technically, you need to refer to what sort of visa your spouse is going to be getting - and thus what sort of visa (or dependent's visa) you'll be moving over on. Most dependents' visas do not allow the "trailing spouse" to work. Period.

Now, that said, you would hardly be the first to do telecommuting while on a non-work visa. It's not actually legal, but I do know people who have gotten away with it.

In theory, you should actually pay UK taxes on the money you earn while overseas and then use the "overseas earned income exclusion" to exclude it from your US taxes. But those I know who have done this take the position that they are doing so to continue their US withholdings (Social Security, health insurance, IRA, etc.) and so continue to pay their US taxes on the money earned. It could raise "warning flags" on your US taxes (i.e. to signal an audit) if your spouse is using the exclusion while you aren't.

But, if your spouse is going over on a company assignment, he may be getting tax assistance -at least to the extent of someone to prepare his tax returns on both sides (you'll still have to file US taxes while you're over there). It might be worthwhile to discuss this idea ahead of time with the tax adviser the company will be using, as there are a number of complications, ranging from the difference in the tax year in the UK to whether or not you will be considered "resident" and/or "domiciled" there during part or all of that 18 months.
Cheers,
Bev