Hello and welcome.
I am NOT a tax professional, just an ordinary ex-pat, now retired, but my first year back in the UK, I had SE income in the US, but no UK income.
You are lucky in that you have almost reached a critical time point which affects your filing situation in the US for 2018. When you meet either the bona fide residence test or the physical presence test, you can either exclude your UK earnings or take a credit for UK tax paid. In any case, you will declare all your earnings for 2018. See
https://www.irs.gov/individuals/international-taxpayers/foreign-earned-income-exclusion-physical-presence-test and have a look at Publication 54
https://www.irs.gov/forms-pubs/about-publication-54.
The UK tax year is April 6-April 5, and if you are on PAYE and have NO foreign income (interest, dividends, employment earnings, rental income or any other type of foreign source income), then you normally never need to file a return here. That part is simple - if your tax code is correct, the PAYE covers your tax due.
Since the US is on a calendar year, you need to do a spreadsheet (or whatever works for you), and calculate your
gross earnings for 2018. Going forward, keep a spreadsheet to make future filing easier. Don't forget any interest you may have earned here, which is normally paid gross (no tax withheld) for most of us. You then convert the UK income to US$. There's no official rate, so you can use a Fx site and calculate the value on the day, or use the US Treasury year-end rate, which is the average for the year. Do both, and see what works to your advantage.
You get an automatic extension to June to file, and can actually extend extend further, but - and this is important - any tax due must still be paid by the normal April 15 deadline. So, figure that part out, and if you owe anything (not forgetting SE tax on your US earnings), make a payment before the deadline if you didn't make estimated payments.
Once you have reached the magic 330 day mark, you can file and exclude your UK earnings; it's still reported, but excluded. See
https://www.irs.gov/individuals/international-taxpayers/foreign-earned-income-exclusion for more detail. This date is also key in claiming the exclusion from health insurance coverage for 2018 (I'm assuming you dodn't continue any cover you may have had after you moved here).
Assuming your wife had no US income, which I suspect is likely, you will file MFS, because you do NOT want to include her income, which you would if MFJ. For most of us, MFS works best, and is far simpler.
If you had an interest in, or signature authority over any non-US financial accounts (bank accounts are the most common) and the aggregate value exceeded $10,000 at any point during the year, you will also need to file a FBAR. That's only done electronically, and it not a tax document; it;s a report to the Treasury Department.
https://bsaefiling.fincen.treas.gov/NoRegFBARFiler.html.
Normally income earned BEFORE you moved to the UK is not taxed in the UK. That would include US income earned from April 6-May 28 when you arrived in the UK.
Others with more and likely better information will likely jump in to help, so don't be shy about asking questions. On the face of it, your situation is relatively simple, and likely only requires a tolerance for paperwork.
A lot of folks use TurboTax or similar software to file their US return, and that may also well work for you.
good luck.
