No, of course they can't force you to wear a face mask.
However, as the other answers have pointed out, they absolutely have a right to require this to work at their company, and if you choose not to or even make ripples against wearing one, expect to be soon finding another job.
Yes, they can. And it has nothing to do with covid 19. Your employer can make you wear whatever they consider a good fit for the job. That can be a branded shirt, or a uniform, or a special hat, or just suit and tie. They can require safety equipment even though the law says you could do without.
The only thing an employer cannot do is require clothing ...
As you were employed after the 6th of April 2012, and haven't been dismissed due to your political opinions or affiliation, and have been employed for less than 2 years, you do not have any grounds to claim unfair dismissal.
Does this mean, regardless of my situation with probation, I was
always in danger of being let go at a weeks notice?
Yup, that is what your contract states in no uncertain terms, and it's reflecting the statutory minimums for notice period in the UK as per https://www.gov.uk/redundancy-your-rights/notice-periods. The company can offer you more, but they ...