Using a solicitor to seek and obtain probate is totally pointless anyway (unless, possibly, the estate is very complex). You can do it without a solicitor.
The oath is possibly most often sworn (or an affirmation given) at a Probate Registry, rather than with a solicitor, as that provides an opportunity for any problems with the application to be ironed out first. (So you don't even need a solicitor for that part of the task).
Where there are two or more executors, the oath can be sworn (or an affirmation given) at different venues but it might slow the probate process a little. If an executor is unable to attend a Probate Registry he/she can swear an oath (or 'affirm') with any solicitor for a total fee of £7. It's not necessary for different executors to use the same solicitor. (Indeed, a solicitor doesn't even need to know that other executors exist. The person swearing/affirming is simply stating "To the best my knowledge, everything on this form is true"; the solicitor isn't required to read what's on the form or to know anything else about the application for a 'grant of representation').