Not really as many troops as you think...
Remember Dunkirk had just taken place and the British army was in a shambles
"The loss of so much materiel on the beaches meant that the British Army needed months to re-supply properly and some planned introductions of new equipment were halted while industrial resources concentrated on making good the losses. Officers told troops falling back from Dunkirk to burn or otherwise disable their trucks (so as not to let them benefit the advancing German forces). The shortage of army vehicles after Dunkirk was so severe that the Royal Army Service Corps (RASC) was reduced to retrieving and refurbishing numbers of obsolete bus and coach models from UK scrapyards to press them into use as troop transports. Some of these antique workhorses were still in use as late as the North African campaign of 1942"
Sec State for War dated June 18th 1940 -
The number of men in the Army at Home today, including
Dominion troops is about 1,313,000, made up as follows:-
Field Army Troops ex B.E.F.- 275,000
" " " in U.K. 320,000
Air Defence 151,000
Coast Defence 13,000
Home Defence Battalions 42,6000
Holding Battalions (under disposal instructions to make up Field Force Units) 49,000
Training Units (half to make up Field Force Units) 365,000
Misc. Establishments 59,400
Canadians 22,000
Australians and New Zeaianders 16,000
Grand Total 1,313,000*
The average monthly intake is just under 50,000 under the
National Service Act, and about 27,000 volunteers.
* '
Note. From the point of view of immediate use it must be
realised that apart from the 90,000 about to be called
up, Air Defence of Great Britain and coast defence
amount to 164,000 while some 150,000 of the grand
total have less than two months service. The total
figure includes 45,000 R.A.M.C. , 45,000 RAOC*,
and 130,000 R.A.S.C , who are not trained to fight.
I think based on these figures (not my own research) it would be unlikely the British army would have been in any state to repel even a smallish (300 000) strong Axis force.
Obviously it all depended on the Channel being Royal Navy free
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