The personal allowance is not available if you elect the remittance basis, so you can only claim it if you file on the arising basis for the year of arrival - in ther words report worldwide income and gains on your UK return.
Actually, HMRC state that for the tax year you arrive in the UK, they normally split it (quite sensibly) into resident and non-resident periods, and you don't need to report anything earned while you were non-resident, as long as you are coming to live in the UK for at least two years.
For the resident period, you do need to pay UK tax. You will most likely be considered as resident and ordinarily resident in the UK, but domiciled in the US, assuming that you intend to go back there at some point (although domicile is a complicated topic as even HMRC admit). Because you are domiciled abroad you can elect either the arising basis or remittance basis for your UK tax.
If you choose the arising basis for the year you arrive, you would still enjoy the full personal allowance even though you are only here part of the tax year. I don't know whether they would tax your bonus though. Best to just ring up HMRC and ask.
If you choose the remittance basis for this year, or any other tax year, and your non-UK income that you leave outside the UK is under £2,000, then you can also keep your full personal allowance. If it's more than that, you not only lose your personal allowance but usually have to pay the £30,000 Remittance Basis Charge.
When you fill in your P46 you should indicate that this is your first job for the tax year.
For more info see the following HMRC link:
http://www.hmrc.gov.uk/incometax/tax-arrive-uk.htmHope this helps!