Generally speaking, granting rights to one bank account (e.g. making a joint account) does not extend rights to other accounts or otherwise let one joint owner create new obligations on the other owner (e.g. opening a line of credit that the other owner must pay for), except to the extent of the joint account. I assume there are no UK rules that would change this feature.
The other party can of course withdraw all the money without need for your approval. This also means that the joint account could be exposed to all the creditors of either party. If your account joint tenant has huge debts, the creditors could theoretically look to the joint account for satisfaction. At least, that would be an issue under US law.
Frankly, it may be simpler to get a separate account for the other person (if possible) and make transfers with online banking. It could also make sense to get a rechargeable banking card, if those are in the UK, which works like a debit card and can be reloaded through various means (sometimes a call, sometimes online deposits, sometimes in physical stores). There may be fees to getting such a card or a second account, of course. The benefit is that the cardholder has no access to your account and you control recharging. Such cards are widely available in the US to people who otherwise would not qualify for traditional bank accounts.
Note also the FATCA complication with adding a US person to your account. My understanding is that a number of non-US banks will simply close the accounts of Americans, rather than deal with FFI hassles under FATCA.